Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Approximate smoothness in normed linear spaces

Approximate smoothness in normed linear spaces We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approx- imate smoothness for some spaces. As an application, we consider in particular the Birkhoff–James orthogonality and its right-additivity under the assumption of approx- imate smoothness. Keywords Smoothness · Rotundity · Approximate smoothness · Approximate rotundity · Birkhoff–James orthogonality · Approximate Birkhoff–James orthogonality · Supporting hyperplanes · Polyhedral spaces · Direct sums Mathematics Subject Classification 46B20; 47L05 · 51F20 · 52B11 1 Introduction Smoothness is definitely one of the most important geometrical properties of normed linear spaces (cf. monographs [7, 13, 15] or a survey [10] for example). In particular, some natural attributes of the Birkhoff–James orthogonality relation can be derived Communicated by Antonio M. Peralta. B Jacek Chmielinski ´ jacek.chmielinski@up.krakow.pl Divya Khurana divyakhurana11@gmail.com ; divya.khurana@iimranchi.ac.in Debmalya Sain saindebmalya@gmail.com Department of Mathematics, Pedagogical University of Krakow, Podchora˙ ¸zych 2, 30-084 Kraków, Poland Humanities and Applied Sciences, IIM Ranchi Suchana Bhawan, Audrey House Campus, Meur’s Road, Ranchi, Jharkhand 834008, India Department of Mathematics, Indian Institute of Information Technology, Raichur, Karnataka 584135, India 0123456789().: V,-vol 41 Page 2 of 23 J. Chmieliński et al. for smooth spaces. Since in its full strength, smoothness can be sometimes a too much restrictive assumption, we are going to propose somehow relaxed, approximate version of this property. We study this concept in general, as well as for some particular linear normed spaces, polyhedral Banach spaces and direct sums of normed linear spaces. 1.1 Notations Throughout the text, we use the symbols X , Y , Z to denote real normed linear spaces. The zero vector of a normed linear space is denoted by θ, but in case of the scalar field R, we simply use the symbol 0. By B := {x ∈ X :x≤ 1} and S := X X {x ∈ X :x= 1}, we denote the unit ball and the unit sphere of X, respectively. The collection of all extreme points of B will be denoted as Ext B .For A ⊂ X, X X diam A := sup x − y denotes the diameter of A. x ,y∈A Let X denote the dual space of X.Given x ∈ X  {θ }, a functional f ∈ S is said to be a supporting functional at x if f (x ) =x . The collection of all supporting functionals at x will be denoted by J (x ), i.e., J (x ) := { f ∈ S : f (x ) =x }, x ∈ X  {θ }. The Hahn–Banach theorem guarantees that the set J (x ) is always nonempty and it is easy to see that it is also convex. It is also known that J (x ) is w -compact. An element x ∈ X  {θ } is said to be a smooth point if J (x ) is a singleton (i.e., J (x ) ={ f } for a unique f ∈ S ). A normed linear space X is called smooth if every x ∈ S (hence, X X every x ∈ X  {θ }) is a smooth point. For f ∈ X  {θ }, we write M := {x ∈ S :| f (x )|= f } f X and M := {x ∈ S : f (x ) = f }. Given two elements x , y ∈ X,let xy:=conv {x , y}={(1 − t )x + ty : t ∈[0, 1]} denote the closed line segment joining x and y.By R(X ) we will denote the length of the “longest” line segment lying on a unit sphere (cf. [5, 20]); more precisely, R(X ) := sup{x − y: xy ⊂ S }. By a hyperplane, we mean a set H :={x ∈ X : f (x ) = c}, Approximate smoothness in normed linear spaces Page 3 of 23 41 where f ∈ X  {θ } is a functional and c ∈ R a constant. Each hyperplane H ⊂ X divides X into two closed half-spaces whose intersection is H itself. We call H a supporting hyperplane to the unit ball,if B lies entirely within one of the two half- spaces and H ∩ B =∅. Equivalently, H is a supporting hyperplane for B if and X X only if there exists f ∈ S such that H ={x ∈ X : f (x ) = 1}. Notice that then we have H ∩ S = M . By H(X ), we denote the set of all supporting hyperplanes for the unit ball in X. Now, we introduce the notion S(X ):= sup{diam (H ∩ S ) : H ∈ H(X )} and by previous observations, we have S(X ) = sup{diam M : f ∈ S ∗ }. (1.1) Actually, using the fact that the diameter of a convex set is the supremum of lengths of segments inside the set and both H ∩ S (for H ∈ H(X )) and M are convex sets, it follows that S(X ) = R(X ). (1.2) We define d : X  {θ}−→[0, 2] by d(x ) = diam J (x ). We also set E (X ):= sup{d(x ) : x ∈ S }. ∗∗ By  : X −→ X , we denote the canonical embedding of a normed linear space ∗∗ X into its bidual X , that is (x )( f ):= f (x ), x ∈ X , f ∈ X . ∗∗ It is known that  is a linear isometry and if it is surjective ((X ) = X ), then X is called a reflexive space (necessarily a Banach space). Using the observation that (M ) = J ( f ) ∩ (X ) and the fact that  is an isometry, we notice that for any normed linear space X: ∗ ∗ E (X ) ≤ S(X ) and S(X ) ≤ E (X ). (1.3) In particular, we have ∗ ∗∗ E (X ) ≤ S(X ) ≤ E (X ). (1.4) An n-dimensional Banach space X is said to be a polyhedral Banach space if B contains only finitely many extreme points or, equivalently, if S is a polyhedron. For more details on polyhedral Banach spaces see [17]. We define the sign function on R by sgn t := , t ∈ R {0} and sgn 0 := 0. |t | 41 Page 4 of 23 J. Chmieliński et al. Let (X , · ) and (Y , · ) be normed linear spaces. Then, X Y X ⊕ Y := {(x , y) : x ∈ X , y ∈ Y }, 1 ≤ p ≤∞, is a normed linear space with respect to the following norm: p p (x  +y ) if p < ∞ X Y (x , y) := max{x  , y } if p =∞. X Y 1 1 Let 1 ≤ q ≤∞ be conjugated to p, i.e., such that + = 1for 1 < p < ∞, q = 1 p q for p =∞ and q =∞ for p = 1. Then, the dual space of X ⊕ Y can be isometrically ∗ ∗ ∗ identified with X ⊕ Y in the following sense: for each F ∈ (X ⊕ Y ) there exist a q p ∗ ∗ unique ( f , g) ∈ X ⊕ Y such that F=( f , g) and F ((x , y)) = f (x ) + g(y) q q for all (x , y) ∈ X ⊕ Y (compare [13,p.5]or[8, Definition 0.18 and Proposition 0.19]). 1.2 Norm derivatives It is well known that smoothness of x ∈ S is equivalent to the Gâteaux differentiability of the norm at x. The concept of norm derivatives arises naturally from the two-sided limiting nature of the Gâteaux derivative of the norm. Let us recall the following definition and basic properties of one-sided norm derivatives. Definition 1.1 Let X be a normed linear space and x , y ∈ X. The norm derivatives of x in the direction of y are defined as 2 2 x + λy−x  x + λy −x ρ (x , y) := x  lim = lim , + + λ→0 λ λ→0 2λ 2 2 x + λy−x  x + λy −x ρ (x , y) := x  lim = lim . − − λ→0 λ λ→0 2λ The following properties of norm derivatives will be useful in this note (see [1] and [9] for proofs): (ρ-i) For all x, y ∈ X and all α ∈ R, αρ (x , y) if α ≥ 0 ρ (αx , y) = ρ (x,αy) = ± ± αρ (x , y) if α< 0. (ρ-ii) ρ (x , y) ≤ ρ (x , y). Moreover, x ∈ X  {θ } is smooth if and only if − + ρ (x , y) = ρ (x , y) for all y ∈ X. − + (ρ-iii) ρ (x , y) =x  sup{ f (y) : f ∈ J (x )}. (ρ-iv) ρ (x , y) =x  inf{ f (y) : f ∈ J (x )}. − Approximate smoothness in normed linear spaces Page 5 of 23 41 1.3 Birkhoff–James orthogonality For vectors x and y in a normed space X, we say that x is Birkhoff–James orthogonal (BJ-orthogonal) to y, written as x ⊥ y,if x + λy≥x , for all λ ∈ R (cf. [2, 11, 12]). James in [11, Theorem 2.1] proved that if x ∈ X  {θ } and y ∈ X, then x ⊥ y if and only if there exists f ∈ J (x ) such that f (y) = 0. Chmielinski ´ in [3] defined an approximate Birkhoff–James orthogonality as fol- lows. Given x , y ∈ X and ε ∈[0, 1), x is said to be approximately orthogonal to y, written as x ⊥ y,if 2 2 x + λy ≥x  − 2εx λy, for all λ ∈ R. (1.5) As shown in [4], condition (1.5) can be equivalently written in the form: x + λy≥x− ελy, for all λ ∈ R. (1.6) In [6, Theorems 2.2 and 2.3], two important characterizations of the considered approx- imate orthogonality were given: x ⊥ y ⇐⇒ ∃ z ∈ span{x , y}: x ⊥ z, z − y≤ εy; (1.7) ⇐⇒ ∃ f ∈ J (x ) :| f (y)|≤ εy. (1.8) Obviously, for any f in J (x ),wehave | f (y)|≤y.If | f (y)| < y then there exists ε ∈[0, 1) such that | f (y)|= εy whence x ⊥ y. In particular, it follows from (1.8)for x , y ∈ S : x ⊥ y for some ε ∈[0, 1) ⇐⇒ y ∈ / M for some f ∈ J (x ) (1.9) or equivalently x ⊥ y for all ε ∈[0, 1) ⇐⇒ y ∈ M for all f ∈ J (x ). (1.10) The study of approximate orthogonality has been an active area of research in recent times, particularly in the space of bounded linear operators on a Banach space (see [6, 14, 16, 18] for recent study on approximate orthogonality). Since Birkhoff–James orthogonality is closely related to the classical notion of smoothness in a normed space, the above works motivated us to introduce a suitable notion of approximate smooth- ness in normed spaces which will be in some sense compatible with approximate orthogonality. In [11], James obtained a characterization of smooth points in terms of right- additivity of the Birkhoff–James orthogonality relation. Namely, x ∈ X  {θ } is a smooth point in X if and only if ⊥ is right-additive at x, that is for any y, z ∈ X: B 41 Page 6 of 23 J. Chmieliński et al. x ⊥ y and x ⊥ z ⇒ x ⊥ (y + z). B B B We now define the notion of an approximate right-additivity of the BJ-orthogonality. Definition 1.2 Let X be a normed linear space and x ∈ X  {θ }. We say that the BJ-orthogonality is approximately right-additive at x (ε-right-additive for some ε ∈ [0, 1)) if for any y, z ∈ X: x ⊥ y and x ⊥ z ⇒ x ⊥ (y + z). B B Similarly, we define the right-additivity of the approximate BJ-orthogonality. Definition 1.3 We say that the approximate BJ-orthogonality is right-additive at x if ε ε whenever x ⊥ y, x ⊥ z for some y, z ∈ X and ε ∈[0, 1), then there exists some B B ε ∈[0, 1) such that x ⊥ (y + z). Observe that in Definition 1.2 for a given x ∈ X  {θ }, ε is uniform in the sense that it is independent on the choice of y and z and in the Definition 1.3 for given x , y, z and ε, ε may depend on all of them. The connections between smoothness and right-additivity of the BJ-orthogonality induce a natural question on a condition, weaker than smoothness, which would char- acterize the approximate right-additivity of the BJ-orthogonality (or right-additivity of the approximate BJ-orthogonality). This is one of the motivations that lead to the notion of approximate smoothness which is stated in the next section. 2 Approximate smoothness and rotundity 2.1 Approximate smoothness—definition and basic properties It is obvious that for any x ∈ X  {θ },0 ≤ d(x ) ≤ 2 and d(x ) = 0 if and only if x is a smooth point. We will be considering the case when the set J (x ) is not necessarily a singleton, but its diameter is limited, in particular strictly less than 2. Definition 2.1 Let X be a normed linear space, x ∈ X  {θ } and ε ∈[0, 2).Wesay that x is ε-smooth if d(x ) ≤ ε. When the value of ε is not specified, we say that x is approximately smooth. The space X is said to be approximately smooth (ε-smooth) if each x ∈ S is ε -smooth for some ε ≤ ε< 2. X x x Observe that in the above definition, we intentionally excluded the possibility of ε = 2. Otherwise, every non-zero element of a normed space would be approxi- mately smooth and every normed space—approximately smooth one. However, our motivation is to distinguish approximately smooth points and spaces, so we restrict to ε< 2. The following lemma will be useful for obtaining a characterization of approximate smoothness in terms of norm derivatives. Approximate smoothness in normed linear spaces Page 7 of 23 41 Lemma 2.2 Let X be a normed linear space, x ∈ X  {θ }. Then, sup {ρ (x , y) − ρ (x , y)}= d(x )x . + − y∈S Proof Using the properties (ρ-iii) and (ρ-iv), for an arbitrary y ∈ S ,wehave ρ (x , y) − ρ (x , y) =x { sup f (y) − inf g(y)} + − g∈J (x ) f ∈J (x ) =x  sup { f (y) − g(y)}. f ,g∈J (x ) Hence, sup {ρ (x , y) − ρ (x , y)}= d(x )x . + − y∈S We now prove the said characterization. Lemma 2.3 Let X be a normed linear space, x ∈ X  {θ } and ε ∈[0, 2). Then, the following conditions are equivalent: (i) x is ε-smooth. (ii) sup {ρ (x , y) − ρ (x , y)}≤ εx . + − y∈S Proof (i) ⇒ (ii) follows from Lemma 2.2. Now, we prove (ii) ⇒ (i). Let f , g ∈ J (x ) and y ∈ S . Then, ρ (x , y) ≥ x  f (y), ρ (x , y) ≥x g(y), ρ (x , y) ≤x  f (y) and ρ (x , y) ≤x g(y). + − − Thus, sup {ρ (x , y) − ρ (x , y)}≥x  sup | f (y) − g(y)|=x  f − g. + − y∈S y∈S X X Hence, (ii) yields x f − g≤ εx  and this proves (i). As an immediate application of Lemma 2.3, we obtain a characterization of the approximate smoothness of the whole space. Corollary 2.4 Let X be a normed linear space and ε ∈[0, 2). Then, the following conditions are equivalent: (i) X is ε-smooth. (ii) sup {ρ (x , y) − ρ (x , y)}≤ ε. + − x ,y∈S If X is a reflexive Banach space then we notice that (M ) = J ( f ) for f ∈ ∗ ∗ X  {θ }. Since  is an isometry so it follows that f ∈ X  {θ } is ε-smooth if and only if diam M ≤ ε. Also, for a reflexive Banach space X using (1.4) and ∗∗ E (X ) = E (X ), we get X is ε-smooth if and only if S(X ) ≤ ε. (2.1) 41 Page 8 of 23 J. Chmieliński et al. Now, we consider a finite-dimensional space. We show that if each vector on a unit sphere is approximately smooth (not necessarily with the same approximation con- stant), then so is the whole space. Basically, this result is a consequence of Weierstrass compactness theorem. Theorem 2.5 Let X be a finite-dimensional Banach space such that each x ∈ S is ε -smooth for some ε ∈[0, 2). Then, X is approximately smooth. x x Proof Let x ∈ S . Without loss of generality, we assume that ε = d(x ). X x Let ε := sup ε . Obviously, ε ≤ 2 and suppose, contrary to our claim, that x ∈S ε = 2. Then, we could find {x }⊂ S such that ε  2. Also, for each n ∈ N,we n X x can choose f , g ∈ J (x ) such that  f − g  >ε − . n n n n n x Now, using the compactness of S , S and the fact that A ={(x , f , g) : X X ∗ ∗ x ∈ S , f , g ∈ J (x )} is a closed set in S × S × S , we can find a con- X X X X vergent subsequence of {(x , f , g )} which we again denote by {(x , f , g )}.Let n n n n n n ∗ ∗ (x , f , g) ∈ A be the limit of {(x , f , g )}.Let h : X × X × X −→ R be n n n defined by h((x , f , g)) = f − g. Then, h is clearly a continuous function. Thus, lim h((x , f , g )) = h((x , f , g)) and this implies that  f − g= 2. This n→∞ n n n contradicts our assumption that ε ∈[0, 2), whence ε< 2 and X is ε-smooth. A normed space is smooth if and only each of its two-dimensional subspaces is smooth (cf. [15, Proposition 5.4.21]). An analogous result can be proved for approx- imate smoothness. However, if the approximation constant ε is not fixed, we can prove it merely for finite-dimensional spaces. The authors do not know whether this assumption is essential. Theorem 2.6 (i) Let X be a normed linear space and ε ∈[0, 2). Then, X is ε-smooth if and only if each of its two-dimensional subspaces is ε-smooth. (ii) Let X be a normed linear space. If X is approximately smooth then so is each of its two-dimensional subspaces. Moreover, if the dimension of X is finite and each of its two-dimensional subspaces is approximately smooth then X is approximately smooth. Proof (i). Let X be ε-smooth space and Y its two-dimensional subspace. For y ∈ Y {θ } any two supporting functionals at y, f , g ˜ ∈ S can be extended (by the Hahn- Banach theorem) to f , g ∈ S —supporting functionals at y in X.Now,  f −˜ g≤ f − g≤ ε implies that Y is ε-smooth. Conversely, let every two-dimensional subspace of X be ε-smooth. Suppose, on the contrary, that X is not ε-smooth. Then, there exist x ∈ S and f , g ∈ J (x ) such that  f − g >ε.Let y ∈ S be such that |( f − g)(y)| >ε. Clearly, this choice of y implies that x and y are linearly independent. Let Y = span{x , y}, f = f | and ˜ ˜ g ˜ = g| . Then, f (x ) = 1, g ˜(x ) = 1 implies that f , g ˜ ∈ S ∗.Also, |( f − g)(y)| >ε Y Y ˜ ˜ ˜ implies that  f −˜ g >ε. Thus, f , g ˜ ∈{h ∈ S ∗ : h(x ) = 1} and  f −˜ g >ε.This leads to the contradiction with ε-smoothness of Y and thus X is ε-smooth. (ii). If X is approximately smooth then, as above, an application of the Hahn– Banach theorem yields approximate smoothness of any two-dimensional subspace of X. Approximate smoothness in normed linear spaces Page 9 of 23 41 Now, we assume that X is a finite-dimensional space and let every two-dimensional subspace of X be approximately smooth. Suppose, on the contrary, that X is not approximately smooth. Then, by Theorem 2.5, there exist x ∈ S and f , g ∈ J (x ) such that  f − g= 2. Let y ∈ S be such that |( f − g)(y)|= 2. Clearly, this choice of y implies that x and y are linearly independent. Let Y = span{x , y}, f = f | and g ˜ = g| . Now, using arguments similar to the proof of (i), we arrive at the contradiction with approximate smoothness of Y and thus X is approximately smooth. The following example shows that in some spaces the notions of smoothness and approximate smoothness can coincide. Example 2.7 (a) Observe that if x = (x , x , x ,...) ∈   {θ } then d(x ) = 2if 1 2 3 1 x = 0for some i ∈ N and d(x ) = 0if x = 0 for all i. Using this observation it i 0 i follows that x ∈   {θ } is ε-smooth for ε ∈[0, 2) if and only if x is smooth. (b) If x = (x , x , x ,...) ∈ c  {θ } then d(x ) = 2 if norm of x is attained at 1 2 3 0 more than one coordinate and d(x ) = 0 if norm of x is attained only at one coordinate. Thus, it follows that x ∈ c  {θ } is ε-smooth for ε ∈[0, 2) if and only if x is smooth. 2.2 Approximate rotundity Rotundity (strict convexity) is another important geometrical property of normed spaces. Although rotundity and smoothness are independent properties, they are related to each other. We would like to show that it is so with their approximate counterparts. Let X be a normed linear space and let ε ∈[0, 2). Definition 2.8 We say that X is ε-rotund (or ε-strictly convex) if S(X ) ≤ ε. Obviously, for ε = 0, the above condition means that each functional f ∈ S ∗ supports the unit sphere in at most one point, which is equivalent to rotundity (cf. [15, Theorem 5.1.15, Corollary 5.1.16]). Basedon(1.3) and (1.4), we can easily establish connections between approximate smoothness (rotundity) of a given space and approximate rotundity (smoothness) of its dual. Theorem 2.9 Let X be a normed linear space and let ε ∈[0, 2). 1. If X is ε-smooth, then X is ε-rotund; 2. If X is ε-rotund, then X is ε-smooth; 3. If X is reflexive, then X is ε-smooth if and only if X is ε-rotund and X is ε-rotund if and only if X is ε-smooth. 3 Approximate additivity of the Birkhoff–James orthogonality in approximately smooth spaces As we have reminded, the Birkhoff–James orthogonality is right-additive at smooth points. Although it is no longer true for non-smooth points, we will show that under a 41 Page 10 of 23 J. Chmieliński et al. Fig. 1 Illustration to Example 3.1 relaxed condition of approximate smoothness, right-additivity can be partially main- tained. We will start with an example showing that, in general, approximate smoothness does not imply approximate right-additivity of the Birkhoff–James orthogonality. Example 3.1 Let X = R with a norm generated by B := conv {(1, 1), (0, 1 + δ), (−1, 1), (−1, −1), (0, −1 − δ), (1, −1)} as a unit ball (with δ> 0). Consider the point P = (0, 1 + δ) and functionals δ 1 f (x , y) = x + y, 1+δ 1+δ δ 1 g(x , y) =− x + y. 1+δ 1+δ Notice that f , g ∈ J (P) (we denote by L and L the respective supporting lines) f g and J (P) = conv { f , g} (see Fig. 1). Then, 2δ 2δ diam J (P) = f − g= sup |x|= . 1 + δ 1 + δ (x ,y)∈B Thus, the considered space is ε-smooth at P, with arbitrarily small ε (if only δ is small enough). On the other hand, for R = (1,δ) and R = (−1,δ),wehave 1 2 P⊥ R , P⊥ R B 1 B 2 2δ ε but R + R = (0, 2δ) = λP with λ = hence P⊥  (R + R ) for any ε ∈[0, 1). 1 2 1 2 1+δ B Although approximate smoothness generally does not imply even approximate right-additivity (no matter how small is ε), the following results give some information in particular situations. Approximate smoothness in normed linear spaces Page 11 of 23 41 Theorem 3.2 Let X be a normed linear space and let x ∈ X {θ }. Let y , y ∈ X {θ } 1 2 ε ε be such that x ⊥ y ,x ⊥ y , where ε ∈[0, 1) is such that 1 2 B B 2y + y 1 2 0 ≤ ε< < 1. 3(y +y ) 1 2 If x is ε-smooth, then there exists ε ∈[0, 1) such that x ⊥ (y + y ). 1 1 2 Proof According to (1.8), we can find f , g ∈ J (x ) such that | f (y )|≤ εy  and 1 1 |g(y )|≤ εy .Now,using ε-smoothness of x, we get 2 2 | f (y )|=|( f − g)(y ) + g(y )| 2 2 2 ≤ f − gy +g(y ) 2 2 ≤ εy + εy 2 2 = 2εy . Using similar arguments, we can show that |g(y )|≤ 2εy . Convexity of J (x ) 1 1 1 1 yields f + g ∈ J (x ) and 2 2 1 1 1 f + g (y + y ) ≤ (εy + 2εy + 2εy + εy ) 1 2 1 2 1 2 2 2 2 = ε(y +y ) 1 2 < y + y . 1 2 1 1 Let ε ∈[0, 1) be such that |( f + g)(y + y )|≤ ε y + y . Then, (1.8) implies 1 1 2 1 1 2 2 2 that x ⊥ (y + y ). 1 2 Theorem 3.3 Let X be a normed linear space and let x ∈ X {θ }. Let y , y ∈ X {θ } 1 2 2y +y 1 2 be such that x ⊥ y ,x ⊥ y .If x is ε-smooth, where 0 ≤ ε< ≤ 2, then B 1 B 2 y +y 1 2 there exists ε ∈[0, 1) such that x ⊥ (y + y ). 1 1 2 Proof We find f , g ∈ J (x ) such that f (y ) = 0 and g(y ) = 0. Now, by ε-smoothness 1 2 of x, we get | f (y )|=|( f − g)(y )|≤ εy , 2 2 2 1 1 and similarly |g(y )|≤ εy . Clearly, f + g ∈ J (x ) and 1 1 2 2 1 1 ε f + g (y + y ) ≤ (y +y )< y + y . 1 2 1 2 1 2 2 2 2 1 1 Let ε ∈[0, 1) be such that f + g (y + y ) ≤ ε y + y .Thus,(1.8) implies 1 1 2 1 1 2 2 2 that x ⊥ (y + y ). 1 2 B 41 Page 12 of 23 J. Chmieliński et al. Fig. 2 Illustration to Example 3.5 We now provide an example to show that in Theorems 3.2 and 3.3, approximate smoothness of x cannot be omitted. We will need the following lemma. Lemma 3.4 Let X be a normed linear space and let x ∈ S . Then, the approximate Birkhoff–James orthogonality is not right-additive at x if and only if there exist y , y ∈ 1 2 y +y 1 2 S ,f , f ∈ J (x ) such that y ∈ / M ,y ∈ / M and ∈ M for all f ∈ J (x ). X 1 2 1 f 2 f f 1 2 y +y 1 2 Proof The proof relies on characterizations (1.9) and (1.10). Let x ∈ S and suppose that the approximate Birkhoff–James orthogonality is not right-additive at x, i.e., there ε ε exist y , y ∈ S , ε ∈[0, 1) such that x ⊥ y , x ⊥ y and there does not exist 1 2 X 1 2 B B y +y 1 1 2 any ε ∈[0, 1) such that x ⊥ . Equivalently, we can write that (due to (1.9)) B y +y 1 2 there exist f , f ∈ J (x ) such that y ∈ / M , y ∈ / M as well as (by (1.10)) that 1 2 1 f 2 f 1 2 y +y 1 2 ∈ M for all f ∈ J (x ). y +y 1 2 If x ∈ X  {θ } is ε-smooth and y , y ∈ X satisfy the conditions stated in Theo- 1 2 y +y 1 2 rem 3.2, then there exists f ∈ J (x ) such that ∈ / M . The following example y +y 1 2 shows that if in Theorems 3.2 or 3.3 the assumption of x being ε-smooth is omitted, the results are not true. Example 3.5 Consider X = (R ,  ).Let x , y , y ∈ S , where x = (1, 1), y = ∞ 1 2 X 1 (1,α), y = (α, 1),0 <α < (see Fig. 2A). Then, we can observe, using (1.7), that 2y +y α α 1+α 1 2 x ⊥ y and x ⊥ y . Moreover, 0 <α < = . 1 2 B B 3 3(y +y ) 1 2 y +y 1 2 Now, we have = x ∈ M for all f ∈ J (x ) and, by Lemma 3.4, approximate y +y 1 2 Birkhoff–James orthogonality is not right-additive at x. To justify essentialness of approximate smoothness in Theorem 3.3 take x , z , z ∈ 1 2 S , where x = (1, 1), z = (1, −α), z = (−α, 1),0 <α < (see Fig. 2B). Then, X 1 2 z +z 1 2 x ⊥ z , x ⊥ z and clearly, = x ∈ M for all f ∈ J (x ).Thus,Lemma 3.4 B 1 B 2 f z +z 1 2 implies that Birkhoff–James orthogonality is not approximately right-additive at x. Finally, in this section, we give a result showing that approximate smoothness of a vector x guarantees the approximate right-additivity of the Birkhoff–James orthogo- nality on some restricted set of vectors. Approximate smoothness in normed linear spaces Page 13 of 23 41 Theorem 3.6 Let X be a real normed linear space and let x ∈ X  {θ } be ε-smooth y +y 1 2 with ε ∈[0, 2). Suppose that y , y ∈ X are such that min{y , y } ≤ . 1 2 1 2 If x ⊥ y and x ⊥ y , then B 1 B 2 ε/2 x ⊥ (y + y ). 1 2 Proof Since x ⊥ y and x ⊥ y , there exist f , g ∈ J (x ) such that f (y ) = g(y ) = B 1 B 2 1 2 0. Therefore, f (y ) = ( f − g)(y ) and g(y ) = (g − f )(y ) 2 2 1 1 which, together with the assumed diam J (x ) ≤ ε,gives | f (y )|≤ εy  and |g(y )|≤ εy . 2 2 1 1 y +y 1 2 Suppose that y ≤y . It follows then from the assumption that y ≤ 1 2 1 and |g(y + y )|=|g(y )|≤ εy ≤ y + y . 1 2 1 1 1 2 ε/2 This means that x ⊥ (y + y ). 1 2 Similarly, if y ≤y , we show that | f (y + y )|≤ y + y , which also 2 1 1 2 1 2 gives the assertion. y +y 1 2 Note that the condition min{y , y } ≤ depends both on directions 1 2 and norms of vectors y , y . It holds true, however, regardless of directions, if y ≥ 1 2 1 3y  or y ≥ 3y . 2 2 1 4 Polyhedral spaces Now, we consider a 2-dimensional regular polyhedral Banach space X with 2n extreme points. Regularity here means that all the edges of the unit sphere are of the same length with respect to the Euclidean metric and all the interior angles are of the same measure. For such spaces, we will calculate the values of d(x ) for each x ∈ Ext B and the value of E (X ). Example 4.1 Let X be a 2-dimensional regular polyhedral Banach space with 2n (n ≥ 2) extreme points and let x ∈ Ext B . Then, 2tan if n is even, 2n d(x ) = E (X ) = π (n−1)π 2tan sin if n is odd. 2n 2n Proof If x ∈ S Ext B then clearly d(x ) = 0. Thus, to calculate E (X ) it is sufficient X X to find d(x ) for all x ∈ Ext B . Moreover, regularity and symmetry of S implies that X X d(x ) = d(y) for all x , y ∈ Ext B . X 41 Page 14 of 23 J. Chmieliński et al. Fig. 3 Illustration to Example 4.1 Without loss of generality, we may assume that Ext B ={v : 1 ≤ k ≤ 2n}, X k (k−1)π (k−1)π where v = cos , sin for 1 ≤ k ≤ 2n (see Fig. 3). n n Let f be the unique support functional for the segment v v for k = 1,..., 2n−1 k k k+1 and v v for k = 2n (on Fig. 3 the respective supporting lines are denoted by L 2n 1 f and L ). Some calculations, which will be omitted here, lead to the explicit formula 2n for the value of f at (x , y) ∈ R : (2k − 1)π (2k − 1)π π f ((x , y)) = x cos + y sin sec 2n 2n 2n (compare with a similar result in [19, the proof of Theorem 3.1]). Moreover, f f if 1 < k ≤ 2n, k−1 k J (v ) = f f if k = 1 2n 1 and thus f − f  if 1 < k ≤ 2n, k−1 k d(v ) = f − f  if k = 1. 2n 1 Approximate smoothness in normed linear spaces Page 15 of 23 41 For 1 < k ≤ 2n,wehave (2k − 1)π (2k − 3)π π ( f − f )((x , y)) = x cos − cos sec k k−1 2n 2n 2n (2k − 1)π (2k − 3)π π + y sin − sin sec 2n 2n 2n (k − 1)π π (k − 1)π π π = −2x sin sin + 2y cos sin sec . n 2n n 2n 2n To calculate the norm of f − f (supremum over the unit sphere), we use the k k−1 Krein–Milman theorem and restrict ourselves to extremal points. Thus, we have for 1 < k ≤ 2n f − f = k k−1 (4.1) (k−1)π π (k−1)π π π max −2x sin sin + 2y cos sin sec . (x ,y)∈Ext B n 2n n 2n 2n If n is even then by taking k = + 1in(4.1), we get π π f n − f n = max 2|x | sin sec . 2 2 (x ,y)∈Ext B 2n 2n n n For (x , y) ∈ Ext B , |x|≤ 1. Thus,  f − f = 2tan . X +1 2n 2 2 If n is odd, then by taking k = n + 1in (4.1), we get π π f − f = max 2|y| sin sec . (4.2) n+1 n (x ,y)∈Ext B 2n 2n (n−1)π For (x , y) ∈ Ext B , |y|≤ sin . Thus, 2n (n − 1)π π π π (n − 1)π f − f = 2sin sin sec = 2tan sin . (4.3) n+1 n 2n 2n 2n 2n 2n Now, using (4.2), (4.3), symmetry and regularity of S , we get 2tan if n is even, 2n E (X ) = d(v ) = π (n−1)π 2tan sin if n is odd. 2n 2n As an application of the above, we obtain the following result on approximate smoothness of a 2-dimensional regular polyhedral Banach space X with 2n extreme points. Proposition 4.2 Let X be a 2-dimensional regular polyhedral Banach space with 2n extreme points (n ≥ 2). Then, the following holds true. (i) X is ε-smooth for ε ∈[2tan , 2), if n is even. 2n 41 Page 16 of 23 J. Chmieliński et al. π (n−1)π (ii) X is ε-smooth for ε ∈[2tan sin , 2),ifn ≥ 3 is odd. 2n 2n In the next result, we give formulas for d(x ), x ∈ S and E (X ) for a finite-dimensional polyhedral Banach space X. Note that in case of X being a finite- dimensional polyhedral Banach space, its dual X is also a polyhedral Banach space with finitely many extreme points in B . Theorem 4.3 Let X be a finite-dimensional polyhedral Banach space and let x ∈ S . Then, (i) d(x ) = max{ f − f : f , f ∈ Ext B such that f , f ∈ J (x )}, i j i j X i j + + (ii) E (X ) = max{ f − f : f , f ∈ Ext B such that M ∩ M =∅}. i j i j X f f i j Proof (i) Let x ∈ S and F , F ,..., F be the facets of S which contain X i i i X 1 2 k x.Let f , f ,..., f ∈ Ext B be the unique supporting functionals for the i i i X 1 2 facets F , F ,..., F , respectively. Let f , g ∈ J (x ). Then, f = α f , i i i j i 1 2 k j =1 j k k k g = β f , where 0 ≤ α ,β ≤ 1 and α = 1, β = 1. Now, j i j j j j j =1 j j =1 j =1 k k k f − g= α f − g = α f − α g j i j i j j j j =1 j =1 j =1 k k = α ( f − g) ≤ α  f − g. j i j i j j j =1 j =1 Similar arguments show that f − g≤ β  f − f ≤ max  f − f i  i i i i j j  j 1≤≤k =1 for all 1 ≤ j ≤ k. Thus,  f − g≤ max  f − f  and this proves the result. i i 1≤ j ,≤k (ii) Observe that if y ∈ F for some facet F and f , g ∈ J (y) then f , g ∈ J (x ) for some x ∈ Ext B ∩ F. Thus, to calculate E (X ) it is sufficient to consider x ∈ Ext B . X X Now, (ii) follows from (i) by observing that if f , g ∈ J (x ) for some x ∈ Ext B ∩ F and facet F then M ∩ M =∅. 5 Direct sums Given normed linear spaces X , Y , we study the space Z = X ⊕ Y,1 ≤ p ≤∞, and approximate smoothness of its elements. We start with a description of the set of supporting functionals. Proposition 5.1 Let X , Y be normed linear spaces and let Z = X ⊕ Y with 1 ≤ p ≤ ∞. Approximate smoothness in normed linear spaces Page 17 of 23 41 1. If 1 < p < ∞ and q is conjugated to p, then for any x ∈ X  {θ } and y ∈ Y  {θ }, we have (a) J ((x , y)) = p−1 p−1 x  f y g ∗ ∗ , ∈ S : f ∈ J (x ), g ∈ J (y) , X ⊕ Y 1 1 q p p q p p q (x  +y ) (x  +y ) (5.1) ∗ ∗ (b) J ((x,θ)) ={( f ,θ) ∈ S : f ∈ J (x )}, X ⊕ Y ∗ ∗ (c) J ((θ , y)) ={(θ , g) ∈ S : g ∈ J (y)}. X ⊕ Y 2 If p = 1,x ∈ X  {θ },y ∈ Y  {θ }, then (a) J ((x , y)) ={( f , g) ∈ S ∗ ∗ : f ∈ J (x ), g ∈ J (y)}, X ⊕ Y (b) J ((x,θ)) ={( f , g) ∈ S ∗ ∗ : f ∈ J (x ), g ∈ B ∗ }, X ⊕ Y Y (c) J ((θ , y)) ={( f , g) ∈ S ∗ ∗ : f ∈ B ∗ , g ∈ J (y)}. X ⊕ Y X 3 Let p =∞ and (x , y) ∈ Z  {θ }. (a) If x  > y, then J ((x , y)) ={( f ,θ) ∈ S ∗ ∗ : f ∈ J (x )}. X ⊕ Y (b) If x  < y, then ∗ ∗ J ((x , y)) ={(θ , g) ∈ S : g ∈ J (y)}. X ⊕ Y (c) If x=y, then {(α f ,(1 − α)g) : f ∈ J (x ), g ∈ J (y), 0 ≤ α ≤ 1}⊆ J ((x , y)). Proof (1a).Let F ∈ J ((x , y)), where F = ( f , g) ∈ S ∗ ∗. Then, using Hölder’s X ⊕ Y inequality, we get p p (x  +y ) =(x , y) = F ((x , y)) = f (x ) + g(y) ≤ f x+gy 1 1 q q p p q p ≤ ( f  +g ) (x  +y ) p p = (x  +y ) . This shows that equality holds in Hölder’s inequality and thus p−1 p−1 x  y f = , g= . (5.2) 1 1 p p p p q q (x  +y ) (x  +y ) 41 Page 18 of 23 J. Chmieliński et al. Also, we get f (x ) = f x , g(y) =gy which implies that f = ∈ J (x ) and g ˜ = ∈ J (y). Combining this with (5.2), we get p−1 p−1 x  f y g ˜ F = ( f , g) = , 1 1 p p p p q q (x  +y ) (x  +y ) which shows ⊆ in (5.1). Now, consider p−1 p−1 x  f y g ∗ ∗ F = , ∈ S , X ⊕ Y 1 1 p p p p q q (x  +y ) (x  +y ) where f ∈ J (x ) and g ∈ J (y). Then, (we use that (p − 1)q = p) p p x  +y F ((x , y)) = =(x , y) . p p (x  +y ) Thus, F ∈ J ((x , y)) and this proves ⊇ in (5.1) and finishes the proof of part (a). Parts (b) and (c) follow using the similar reasoning. (2). For the proof of (a) let x ∈ X \{θ }, y ∈ Y \{θ }, f ∈ J (x ) and g ∈ J (y). Then, ∗ ∗ ( f , g) ∈ S and ( f , g)((x , y)) = f (x ) + g(y) =x+y=(x , y) . X ⊕ Y 1 Thus, ( f , g) ∈ J ((x , y)). For the reverse, let ( f , g) ∈ J ((x , y)). Then,  f ≤ 1, g≤ 1 and x+y=(x , y) = ( f , g)((x , y)) = f (x ) + g(y) ≤ f x+gy≤x+y. This shows that  f = 1, g= 1, f (x ) =x  and g(y) =y. Thus, f ∈ J (x ) and g ∈ J (y). ∗ ∗ ∗ To prove (b) let x ∈ X \{θ }, f ∈ J (x ) and g ∈ B . Then, ( f , g) ∈ S and Y X ⊕ Y ( f , g)((x,θ)) = f (x ) =x=(x,θ) . Thus, ( f , g) ∈ J ((x,θ)). The reverse inclusion is clear. The proof of (c) is analogous. (3). For the proof of ((a))let x ∈ X \{θ }, y ∈ Y be such that x  > y.Let f ∈ ∗ ∗ J (x ). Then, clearly ( f ,θ) ∈ S , ( f , θ )((x , y)) = f (x ) =x=(x , y) X ⊕ Y ∞ and thus ( f ,θ) ∈ J ((x , y)). Now, let ( f , g) ∈ J ((x , y)), where g ∈ B .If f = θ, then ( f , g)((x , y)) = g(y) ≤y < (x , y) . Thus, f = θ. Suppose g = θ. Since ( f , g) ∈ S ∗ ∗, f = θ, g = θ,gives  f  < 1 X ⊕ Y and g < 1. Also,  f +g= 1 implies g= 1 − f . Thus, x=(x , y) = ( f , g)((x , y)) = f (x ) + g(y) ≤ f x+gy <  f x+ (1 − f )x=x . Approximate smoothness in normed linear spaces Page 19 of 23 41 The obtained contradiction proves the required form of J ((x , y)). The proof of (b) is analogous. Finally, we prove (c).Let x ∈ X \{θ }, y ∈ Y \{θ } be such that x=y.Let ∗ ∗ f ∈ J (x ), g ∈ J (y) and 0 ≤ α ≤ 1. Then, (α f ,(1 − α)g) ∈ S and (α f ,(1 − X ⊕ Y α)g)((x , y)) = α f (x ) + (1 − α)g(y) = αx+ (1 − α)y=x=(x , y) . This proves the result. Corollary 5.2 Let X , Y be normed linear spaces and let Z = X ⊕ Y with 1 ≤ p ≤∞. 1. If 1 < p < ∞ and q is conjugated to p, then for any x ∈ X  {θ } and y ∈ Y  {θ }, we have (a) p p x  y q q d((x , y)) = d(x ) + d(y) , (5.3) p p p p x  +y x  +y (b) d((x,θ)) = d(x ), (c) d((θ , y)) = d(y). 2. If p = 1,x ∈ X  {θ },y ∈ Y  {θ }, then (a) d((x , y)) = max{d(x ), d(y)}, (b) d((x,θ)) = 2, (c) d((θ , y)) = 2. 3. Let p =∞, (x , y) ∈ Z  {θ }. (a) If x  > y, then d((x , y)) = d(x ). (b) If x  < y, then d((x , y)) = d(y). (c) If x=y, then d((x , y)) = 2. Proof We will prove (1a).Let F , G ∈ J ((x , y)). On account of (5.1), there exist f , g ∈ J (x ), f , g ∈ J (y) such that x x y y p−1 p−1 x  f y f x y F = , 1 1 p p p p q q (x  +y ) (x  +y ) and p−1 p−1 y g x  g x y G = , . 1 1 p p q p p q (x  +y ) (x  +y ) This gives (using (p − 1)q = p) p p x  y q q F − G =  f − g  +  f − g q x x y y p p p p x  +y x  +y 41 Page 20 of 23 J. Chmieliński et al. and thus, p p x  y q q q d((x , y)) ≤ d(x ) + d(y) . p p p p x  +y x  +y δ δ To prove the reverse inequality fix arbitrarily δ> 0 and choose f , g ∈ J (x ) such x x δ δ q q δ δ that  f − g  > d(x ) − δ. Analogously, let f , g ∈ J (y) be chosen such that x x y y δ δ q q f − g  > d(y) − δ. Define y y p−1 δ p−1 δ y f x  f δ x F := , 1 1 p p q p p q (x  +y ) (x  +y ) and p−1 δ p−1 δ y g x  g G := , . 1 1 p p p p q q (x  +y ) (x  +y ) δ δ By (5.1), F , G ∈ J ((x , y)) whence q δ δ d((x , y)) ≥F − G p p x  y q q > (d(x ) − δ) + (d(y) − δ) p p p p x  +y x  +y and since δ> 0 was arbitrary, we get p p x  y q q q d((x , y)) ≥ d(x ) + d(y) . p p p p x  +y x  +y The proofs of other cases are similar or obvious. Now, as a straightforward consequence of Corollary 5.2, we characterize approxi- mate smoothness of the direct sum. Theorem 5.3 Let X, Y be normed linear spaces and Z = X ⊕ Y, 1 < p < ∞. Then, the following statements hold true: (i) Let x ∈ X  {θ },y ∈ Y  {θ }. (a) If x is ε -smooth in X and y is ε -smoothinY for ε ,ε ∈[0, 2), then (x , y) x y x y is ε-smooth in Z with p p x  y q q ε := ε + ε . x y p p p p x  +y x  +y (b) If x is ε-smooth in X and y is ε-smoothinY for ε ∈[0, 2), then (x , y) is ε-smooth in Z. Approximate smoothness in normed linear spaces Page 21 of 23 41 (c) If (x , y) is ε-smooth in Z for ε ∈[0, 2), then either x is ε-smooth in X or y is ε-smoothinY. (d) (x , y) is approximately smooth if and only if either x or y is approximately smooth. (ii) Let x ∈ X  {θ }. Then, (x,θ) ∈ Zis ε-smooth for ε ∈[0, 2) if and only if x is ε-smooth in X. (iii) Let y ∈ Y  {θ }. Then, (θ , y) ∈ Zis ε-smooth for ε ∈[0, 2) if and only if y is ε-smoothinY. Proof For the proof of (i), we need to apply Corollary 5.2 (1a). In particular, for (id), we observe that d(x , y)< 2 if and only if d(x)< 2or d(y)< 2. The statements (ii) and (iii) immediately follow from Corollary 5.2 (1b) and (1c), respectively. Theorem 5.4 Let X, Y be normed linear spaces and Z = X ⊕ Y . Then, the following hold true: (i) (a) If x is ε -smooth in X and y is ε -smoothinY for ε ,ε ∈[0, 2), then (x , y) x y x y is ε-smooth in Z with ε := max{ε ,ε }. x y (b) If x is ε-smooth in X and y is ε-smoothinY for ε ∈[0, 2), then (x , y) is ε-smooth in Z. (c) If (x , y) is ε-smooth in Z for ε ∈[0, 2), then both x and y are ε-smooth in X and Y , respectively. (d) (x , y) is approximately smooth if and only if both x and y are approximately smooth. (ii) If x ∈ X  {θ } then (x,θ) ∈ Z cannot be approximately smooth. (iii) If y ∈ Y  {θ } then (θ , y) ∈ Z cannot be approximately smooth. Proof The proof of (i) relies on Corollary 5.2 (2a). Statements (ii) and (iii) follow immediately from Corollary 5.2 (2b) and (2c), respectively. Theorem 5.5 Let X, Y be normed linear spaces, Z = X ⊕ Y and z = (x , y) ∈ Z  {θ }. Then, the following statements hold true: (i) Let x  > y. Then, z is ε-smooth in Z for ε ∈[0, 2) if and only if x is ε-smooth in X. (ii) Let x  < y. Then, z is ε-smooth in Z for ε ∈[0, 2) if and only if y is ε-smooth in Y . (iii) If x=y, then z cannot be approximately smooth. Proof Statements (i) and (ii) follow from Corollary 5.2 (3a) and (3b), whereas (iii) is a consequence of Corollary 5.2 (3c). The following final example is an application of the above theorem and the fact that any t ∈ R {0} is a smooth point of R. Example 5.6 Let Z = X ⊕ R be a 3-dimensional polyhedral Banach space whose unit ball is a right prism with regular polyhedron P as its base. Then, the following hold true for z = (x , t ) ∈ Z  {θ }: 41 Page 22 of 23 J. Chmieliński et al. (i) Let x  > |t |. Then, z is ε-smooth in Z for ε ∈[0, 2) if and only if x is ε-smooth in X. (ii) Let x  < |t |. Then, z is smooth. (iii) If x=|t |, then z cannot be approximately smooth. Acknowledgements The research of Divya Khurana is sponsored by Dr. D. S. Kothari Postdoctoral Fel- lowship under the mentorship of Professor Gadadhar Misra. She would also like to thank Professor Gideon Schechtman for discussion on the direct sum of normed linear spaces. The research of Dr. Debmalya Sain is sponsored by DST-SERB under the mentorship of Professor Apoorva Khare. Dr. Sain is elated to acknowl- edge the monumental positive role played by the Department of Mathematics, Indian Institute of Science, for providing him with a rich mathematical setting. The authors thank the referees for their valuable comments. Declarations Conflict of interest The authors state no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Alsina, C., Sikorska, J., Santos Tomás, M.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack (2010) 2. Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935) 3. Chmielinski, ´ J.: On an -Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Article 79 (2005) 4. Chmielinski, ´ J.: Approximate Birkhoff–James orthogonality in normed linear spaces and related top- ics. In: Aron, R.M., Moslehian, M.S., Spitkovsky, I.M., Woerdeman, H.J. (eds.) Operator and Norm Inequalities and Related Topics, 303–320. Birkhäuser, Cham (2022) 5. Chmielinski, ´ J., Khurana, D., Sain, D.: Local approximate symmetry of Birkhoff–James orthogonality in normed linear spaces. Results Math. 76, 136 (2021) 6. Chmielinski, ´ J., Stypuła, T., Wójcik, P.: Approximate orthogonality in normed spaces and its applica- tions. Linear Algebra Appl. 531, 305–317 (2017) 7. Day, M.M.: Normed Linear Spaces. Springer, Berlin (1973) 8. Van Dulst, D.: Reflexive and Superreflexive Banach Spaces, Mathematical Centre Tracts 102. Mathe- matisch Centrum, Amsterdam (1978) 9. Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, Hauppauge, NY (2004) 10. Fabian, M., Montesinos, V., Zizler, V.: Smoothness in Banach spaces. Selected problems. Rev. R. Acad. Cienc Exactas Fís. Nat. Ser. A Mat. 100, 101–125 (2006) 11. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947) 12. James, R.C.: Inner product in normed linear spaces. Bull. Am. Math. Soc. 53, 559–566 (1947) 13. Johnson, W.B., Lindenstrauss, J.: Basic concepts in the geometry of Banach spaces. In: Johnson, W.B, Lindenstrauss J. (eds), Handbook of geometry of Banach spaces, vol. 1, North-Holland Publishing Co., Amsterdam, pp. 1–84 (2001) 14. Mal, A., Paul, K., Rao, T.S.S.R.K., Sain, D.: Approximate Birkhoff–James orthogonality and smooth- ness in the space of bounded linear operators. Monatsh. Math. 190(3), 549–558 (2019) Approximate smoothness in normed linear spaces Page 23 of 23 41 15. Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer-Verlag, New York (1998) 16. Paul, K., Sain, D., Mal, A.: Approximate Birkhoff–James orthogonality in the space of bounded linear operators. Linear Algebra Appl. 537, 348–357 (2018) 17. Sain, D., Paul, K., Bhunia, P., Bag, S.: On the numerical index of polyhedral Banach spaces. Linear Algebra Appl. 577, 121–133 (2019) 18. Sain, D., Paul, K., Mal, A.: On approximate Birkhoff–James orthogonality and normal cones in a normed space. J. Convex Anal. 26(1), 341–351 (2019) 19. Sain, D., Roy, S., Bagchi, S., Balestro, V.: A study of symmetric points in Banach spaces. Linear Multilinear Algebra 70(5), 888–898 (2020) 20. Stypuła, T., Wójcik, P.: Characterizations of rotundity and smoothness by approximate orthogonalities. Ann. Math. Sil. 30, 183–201 (2016) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Banach Journal of Mathematical Analysis Springer Journals

Approximate smoothness in normed linear spaces

Download PDF
23 pages

Loading next page...
 
/lp/springer-journals/approximate-smoothness-in-normed-linear-spaces-AZ06vKW9OA

References (35)

Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2023
ISSN
2662-2033
eISSN
1735-8787
DOI
10.1007/s43037-023-00263-4
Publisher site
See Article on Publisher Site

Abstract

We introduce the notion of approximate smoothness in a normed linear space. We characterize this property and show the connections between smoothness and approx- imate smoothness for some spaces. As an application, we consider in particular the Birkhoff–James orthogonality and its right-additivity under the assumption of approx- imate smoothness. Keywords Smoothness · Rotundity · Approximate smoothness · Approximate rotundity · Birkhoff–James orthogonality · Approximate Birkhoff–James orthogonality · Supporting hyperplanes · Polyhedral spaces · Direct sums Mathematics Subject Classification 46B20; 47L05 · 51F20 · 52B11 1 Introduction Smoothness is definitely one of the most important geometrical properties of normed linear spaces (cf. monographs [7, 13, 15] or a survey [10] for example). In particular, some natural attributes of the Birkhoff–James orthogonality relation can be derived Communicated by Antonio M. Peralta. B Jacek Chmielinski ´ jacek.chmielinski@up.krakow.pl Divya Khurana divyakhurana11@gmail.com ; divya.khurana@iimranchi.ac.in Debmalya Sain saindebmalya@gmail.com Department of Mathematics, Pedagogical University of Krakow, Podchora˙ ¸zych 2, 30-084 Kraków, Poland Humanities and Applied Sciences, IIM Ranchi Suchana Bhawan, Audrey House Campus, Meur’s Road, Ranchi, Jharkhand 834008, India Department of Mathematics, Indian Institute of Information Technology, Raichur, Karnataka 584135, India 0123456789().: V,-vol 41 Page 2 of 23 J. Chmieliński et al. for smooth spaces. Since in its full strength, smoothness can be sometimes a too much restrictive assumption, we are going to propose somehow relaxed, approximate version of this property. We study this concept in general, as well as for some particular linear normed spaces, polyhedral Banach spaces and direct sums of normed linear spaces. 1.1 Notations Throughout the text, we use the symbols X , Y , Z to denote real normed linear spaces. The zero vector of a normed linear space is denoted by θ, but in case of the scalar field R, we simply use the symbol 0. By B := {x ∈ X :x≤ 1} and S := X X {x ∈ X :x= 1}, we denote the unit ball and the unit sphere of X, respectively. The collection of all extreme points of B will be denoted as Ext B .For A ⊂ X, X X diam A := sup x − y denotes the diameter of A. x ,y∈A Let X denote the dual space of X.Given x ∈ X  {θ }, a functional f ∈ S is said to be a supporting functional at x if f (x ) =x . The collection of all supporting functionals at x will be denoted by J (x ), i.e., J (x ) := { f ∈ S : f (x ) =x }, x ∈ X  {θ }. The Hahn–Banach theorem guarantees that the set J (x ) is always nonempty and it is easy to see that it is also convex. It is also known that J (x ) is w -compact. An element x ∈ X  {θ } is said to be a smooth point if J (x ) is a singleton (i.e., J (x ) ={ f } for a unique f ∈ S ). A normed linear space X is called smooth if every x ∈ S (hence, X X every x ∈ X  {θ }) is a smooth point. For f ∈ X  {θ }, we write M := {x ∈ S :| f (x )|= f } f X and M := {x ∈ S : f (x ) = f }. Given two elements x , y ∈ X,let xy:=conv {x , y}={(1 − t )x + ty : t ∈[0, 1]} denote the closed line segment joining x and y.By R(X ) we will denote the length of the “longest” line segment lying on a unit sphere (cf. [5, 20]); more precisely, R(X ) := sup{x − y: xy ⊂ S }. By a hyperplane, we mean a set H :={x ∈ X : f (x ) = c}, Approximate smoothness in normed linear spaces Page 3 of 23 41 where f ∈ X  {θ } is a functional and c ∈ R a constant. Each hyperplane H ⊂ X divides X into two closed half-spaces whose intersection is H itself. We call H a supporting hyperplane to the unit ball,if B lies entirely within one of the two half- spaces and H ∩ B =∅. Equivalently, H is a supporting hyperplane for B if and X X only if there exists f ∈ S such that H ={x ∈ X : f (x ) = 1}. Notice that then we have H ∩ S = M . By H(X ), we denote the set of all supporting hyperplanes for the unit ball in X. Now, we introduce the notion S(X ):= sup{diam (H ∩ S ) : H ∈ H(X )} and by previous observations, we have S(X ) = sup{diam M : f ∈ S ∗ }. (1.1) Actually, using the fact that the diameter of a convex set is the supremum of lengths of segments inside the set and both H ∩ S (for H ∈ H(X )) and M are convex sets, it follows that S(X ) = R(X ). (1.2) We define d : X  {θ}−→[0, 2] by d(x ) = diam J (x ). We also set E (X ):= sup{d(x ) : x ∈ S }. ∗∗ By  : X −→ X , we denote the canonical embedding of a normed linear space ∗∗ X into its bidual X , that is (x )( f ):= f (x ), x ∈ X , f ∈ X . ∗∗ It is known that  is a linear isometry and if it is surjective ((X ) = X ), then X is called a reflexive space (necessarily a Banach space). Using the observation that (M ) = J ( f ) ∩ (X ) and the fact that  is an isometry, we notice that for any normed linear space X: ∗ ∗ E (X ) ≤ S(X ) and S(X ) ≤ E (X ). (1.3) In particular, we have ∗ ∗∗ E (X ) ≤ S(X ) ≤ E (X ). (1.4) An n-dimensional Banach space X is said to be a polyhedral Banach space if B contains only finitely many extreme points or, equivalently, if S is a polyhedron. For more details on polyhedral Banach spaces see [17]. We define the sign function on R by sgn t := , t ∈ R {0} and sgn 0 := 0. |t | 41 Page 4 of 23 J. Chmieliński et al. Let (X , · ) and (Y , · ) be normed linear spaces. Then, X Y X ⊕ Y := {(x , y) : x ∈ X , y ∈ Y }, 1 ≤ p ≤∞, is a normed linear space with respect to the following norm: p p (x  +y ) if p < ∞ X Y (x , y) := max{x  , y } if p =∞. X Y 1 1 Let 1 ≤ q ≤∞ be conjugated to p, i.e., such that + = 1for 1 < p < ∞, q = 1 p q for p =∞ and q =∞ for p = 1. Then, the dual space of X ⊕ Y can be isometrically ∗ ∗ ∗ identified with X ⊕ Y in the following sense: for each F ∈ (X ⊕ Y ) there exist a q p ∗ ∗ unique ( f , g) ∈ X ⊕ Y such that F=( f , g) and F ((x , y)) = f (x ) + g(y) q q for all (x , y) ∈ X ⊕ Y (compare [13,p.5]or[8, Definition 0.18 and Proposition 0.19]). 1.2 Norm derivatives It is well known that smoothness of x ∈ S is equivalent to the Gâteaux differentiability of the norm at x. The concept of norm derivatives arises naturally from the two-sided limiting nature of the Gâteaux derivative of the norm. Let us recall the following definition and basic properties of one-sided norm derivatives. Definition 1.1 Let X be a normed linear space and x , y ∈ X. The norm derivatives of x in the direction of y are defined as 2 2 x + λy−x  x + λy −x ρ (x , y) := x  lim = lim , + + λ→0 λ λ→0 2λ 2 2 x + λy−x  x + λy −x ρ (x , y) := x  lim = lim . − − λ→0 λ λ→0 2λ The following properties of norm derivatives will be useful in this note (see [1] and [9] for proofs): (ρ-i) For all x, y ∈ X and all α ∈ R, αρ (x , y) if α ≥ 0 ρ (αx , y) = ρ (x,αy) = ± ± αρ (x , y) if α< 0. (ρ-ii) ρ (x , y) ≤ ρ (x , y). Moreover, x ∈ X  {θ } is smooth if and only if − + ρ (x , y) = ρ (x , y) for all y ∈ X. − + (ρ-iii) ρ (x , y) =x  sup{ f (y) : f ∈ J (x )}. (ρ-iv) ρ (x , y) =x  inf{ f (y) : f ∈ J (x )}. − Approximate smoothness in normed linear spaces Page 5 of 23 41 1.3 Birkhoff–James orthogonality For vectors x and y in a normed space X, we say that x is Birkhoff–James orthogonal (BJ-orthogonal) to y, written as x ⊥ y,if x + λy≥x , for all λ ∈ R (cf. [2, 11, 12]). James in [11, Theorem 2.1] proved that if x ∈ X  {θ } and y ∈ X, then x ⊥ y if and only if there exists f ∈ J (x ) such that f (y) = 0. Chmielinski ´ in [3] defined an approximate Birkhoff–James orthogonality as fol- lows. Given x , y ∈ X and ε ∈[0, 1), x is said to be approximately orthogonal to y, written as x ⊥ y,if 2 2 x + λy ≥x  − 2εx λy, for all λ ∈ R. (1.5) As shown in [4], condition (1.5) can be equivalently written in the form: x + λy≥x− ελy, for all λ ∈ R. (1.6) In [6, Theorems 2.2 and 2.3], two important characterizations of the considered approx- imate orthogonality were given: x ⊥ y ⇐⇒ ∃ z ∈ span{x , y}: x ⊥ z, z − y≤ εy; (1.7) ⇐⇒ ∃ f ∈ J (x ) :| f (y)|≤ εy. (1.8) Obviously, for any f in J (x ),wehave | f (y)|≤y.If | f (y)| < y then there exists ε ∈[0, 1) such that | f (y)|= εy whence x ⊥ y. In particular, it follows from (1.8)for x , y ∈ S : x ⊥ y for some ε ∈[0, 1) ⇐⇒ y ∈ / M for some f ∈ J (x ) (1.9) or equivalently x ⊥ y for all ε ∈[0, 1) ⇐⇒ y ∈ M for all f ∈ J (x ). (1.10) The study of approximate orthogonality has been an active area of research in recent times, particularly in the space of bounded linear operators on a Banach space (see [6, 14, 16, 18] for recent study on approximate orthogonality). Since Birkhoff–James orthogonality is closely related to the classical notion of smoothness in a normed space, the above works motivated us to introduce a suitable notion of approximate smooth- ness in normed spaces which will be in some sense compatible with approximate orthogonality. In [11], James obtained a characterization of smooth points in terms of right- additivity of the Birkhoff–James orthogonality relation. Namely, x ∈ X  {θ } is a smooth point in X if and only if ⊥ is right-additive at x, that is for any y, z ∈ X: B 41 Page 6 of 23 J. Chmieliński et al. x ⊥ y and x ⊥ z ⇒ x ⊥ (y + z). B B B We now define the notion of an approximate right-additivity of the BJ-orthogonality. Definition 1.2 Let X be a normed linear space and x ∈ X  {θ }. We say that the BJ-orthogonality is approximately right-additive at x (ε-right-additive for some ε ∈ [0, 1)) if for any y, z ∈ X: x ⊥ y and x ⊥ z ⇒ x ⊥ (y + z). B B Similarly, we define the right-additivity of the approximate BJ-orthogonality. Definition 1.3 We say that the approximate BJ-orthogonality is right-additive at x if ε ε whenever x ⊥ y, x ⊥ z for some y, z ∈ X and ε ∈[0, 1), then there exists some B B ε ∈[0, 1) such that x ⊥ (y + z). Observe that in Definition 1.2 for a given x ∈ X  {θ }, ε is uniform in the sense that it is independent on the choice of y and z and in the Definition 1.3 for given x , y, z and ε, ε may depend on all of them. The connections between smoothness and right-additivity of the BJ-orthogonality induce a natural question on a condition, weaker than smoothness, which would char- acterize the approximate right-additivity of the BJ-orthogonality (or right-additivity of the approximate BJ-orthogonality). This is one of the motivations that lead to the notion of approximate smoothness which is stated in the next section. 2 Approximate smoothness and rotundity 2.1 Approximate smoothness—definition and basic properties It is obvious that for any x ∈ X  {θ },0 ≤ d(x ) ≤ 2 and d(x ) = 0 if and only if x is a smooth point. We will be considering the case when the set J (x ) is not necessarily a singleton, but its diameter is limited, in particular strictly less than 2. Definition 2.1 Let X be a normed linear space, x ∈ X  {θ } and ε ∈[0, 2).Wesay that x is ε-smooth if d(x ) ≤ ε. When the value of ε is not specified, we say that x is approximately smooth. The space X is said to be approximately smooth (ε-smooth) if each x ∈ S is ε -smooth for some ε ≤ ε< 2. X x x Observe that in the above definition, we intentionally excluded the possibility of ε = 2. Otherwise, every non-zero element of a normed space would be approxi- mately smooth and every normed space—approximately smooth one. However, our motivation is to distinguish approximately smooth points and spaces, so we restrict to ε< 2. The following lemma will be useful for obtaining a characterization of approximate smoothness in terms of norm derivatives. Approximate smoothness in normed linear spaces Page 7 of 23 41 Lemma 2.2 Let X be a normed linear space, x ∈ X  {θ }. Then, sup {ρ (x , y) − ρ (x , y)}= d(x )x . + − y∈S Proof Using the properties (ρ-iii) and (ρ-iv), for an arbitrary y ∈ S ,wehave ρ (x , y) − ρ (x , y) =x { sup f (y) − inf g(y)} + − g∈J (x ) f ∈J (x ) =x  sup { f (y) − g(y)}. f ,g∈J (x ) Hence, sup {ρ (x , y) − ρ (x , y)}= d(x )x . + − y∈S We now prove the said characterization. Lemma 2.3 Let X be a normed linear space, x ∈ X  {θ } and ε ∈[0, 2). Then, the following conditions are equivalent: (i) x is ε-smooth. (ii) sup {ρ (x , y) − ρ (x , y)}≤ εx . + − y∈S Proof (i) ⇒ (ii) follows from Lemma 2.2. Now, we prove (ii) ⇒ (i). Let f , g ∈ J (x ) and y ∈ S . Then, ρ (x , y) ≥ x  f (y), ρ (x , y) ≥x g(y), ρ (x , y) ≤x  f (y) and ρ (x , y) ≤x g(y). + − − Thus, sup {ρ (x , y) − ρ (x , y)}≥x  sup | f (y) − g(y)|=x  f − g. + − y∈S y∈S X X Hence, (ii) yields x f − g≤ εx  and this proves (i). As an immediate application of Lemma 2.3, we obtain a characterization of the approximate smoothness of the whole space. Corollary 2.4 Let X be a normed linear space and ε ∈[0, 2). Then, the following conditions are equivalent: (i) X is ε-smooth. (ii) sup {ρ (x , y) − ρ (x , y)}≤ ε. + − x ,y∈S If X is a reflexive Banach space then we notice that (M ) = J ( f ) for f ∈ ∗ ∗ X  {θ }. Since  is an isometry so it follows that f ∈ X  {θ } is ε-smooth if and only if diam M ≤ ε. Also, for a reflexive Banach space X using (1.4) and ∗∗ E (X ) = E (X ), we get X is ε-smooth if and only if S(X ) ≤ ε. (2.1) 41 Page 8 of 23 J. Chmieliński et al. Now, we consider a finite-dimensional space. We show that if each vector on a unit sphere is approximately smooth (not necessarily with the same approximation con- stant), then so is the whole space. Basically, this result is a consequence of Weierstrass compactness theorem. Theorem 2.5 Let X be a finite-dimensional Banach space such that each x ∈ S is ε -smooth for some ε ∈[0, 2). Then, X is approximately smooth. x x Proof Let x ∈ S . Without loss of generality, we assume that ε = d(x ). X x Let ε := sup ε . Obviously, ε ≤ 2 and suppose, contrary to our claim, that x ∈S ε = 2. Then, we could find {x }⊂ S such that ε  2. Also, for each n ∈ N,we n X x can choose f , g ∈ J (x ) such that  f − g  >ε − . n n n n n x Now, using the compactness of S , S and the fact that A ={(x , f , g) : X X ∗ ∗ x ∈ S , f , g ∈ J (x )} is a closed set in S × S × S , we can find a con- X X X X vergent subsequence of {(x , f , g )} which we again denote by {(x , f , g )}.Let n n n n n n ∗ ∗ (x , f , g) ∈ A be the limit of {(x , f , g )}.Let h : X × X × X −→ R be n n n defined by h((x , f , g)) = f − g. Then, h is clearly a continuous function. Thus, lim h((x , f , g )) = h((x , f , g)) and this implies that  f − g= 2. This n→∞ n n n contradicts our assumption that ε ∈[0, 2), whence ε< 2 and X is ε-smooth. A normed space is smooth if and only each of its two-dimensional subspaces is smooth (cf. [15, Proposition 5.4.21]). An analogous result can be proved for approx- imate smoothness. However, if the approximation constant ε is not fixed, we can prove it merely for finite-dimensional spaces. The authors do not know whether this assumption is essential. Theorem 2.6 (i) Let X be a normed linear space and ε ∈[0, 2). Then, X is ε-smooth if and only if each of its two-dimensional subspaces is ε-smooth. (ii) Let X be a normed linear space. If X is approximately smooth then so is each of its two-dimensional subspaces. Moreover, if the dimension of X is finite and each of its two-dimensional subspaces is approximately smooth then X is approximately smooth. Proof (i). Let X be ε-smooth space and Y its two-dimensional subspace. For y ∈ Y {θ } any two supporting functionals at y, f , g ˜ ∈ S can be extended (by the Hahn- Banach theorem) to f , g ∈ S —supporting functionals at y in X.Now,  f −˜ g≤ f − g≤ ε implies that Y is ε-smooth. Conversely, let every two-dimensional subspace of X be ε-smooth. Suppose, on the contrary, that X is not ε-smooth. Then, there exist x ∈ S and f , g ∈ J (x ) such that  f − g >ε.Let y ∈ S be such that |( f − g)(y)| >ε. Clearly, this choice of y implies that x and y are linearly independent. Let Y = span{x , y}, f = f | and ˜ ˜ g ˜ = g| . Then, f (x ) = 1, g ˜(x ) = 1 implies that f , g ˜ ∈ S ∗.Also, |( f − g)(y)| >ε Y Y ˜ ˜ ˜ implies that  f −˜ g >ε. Thus, f , g ˜ ∈{h ∈ S ∗ : h(x ) = 1} and  f −˜ g >ε.This leads to the contradiction with ε-smoothness of Y and thus X is ε-smooth. (ii). If X is approximately smooth then, as above, an application of the Hahn– Banach theorem yields approximate smoothness of any two-dimensional subspace of X. Approximate smoothness in normed linear spaces Page 9 of 23 41 Now, we assume that X is a finite-dimensional space and let every two-dimensional subspace of X be approximately smooth. Suppose, on the contrary, that X is not approximately smooth. Then, by Theorem 2.5, there exist x ∈ S and f , g ∈ J (x ) such that  f − g= 2. Let y ∈ S be such that |( f − g)(y)|= 2. Clearly, this choice of y implies that x and y are linearly independent. Let Y = span{x , y}, f = f | and g ˜ = g| . Now, using arguments similar to the proof of (i), we arrive at the contradiction with approximate smoothness of Y and thus X is approximately smooth. The following example shows that in some spaces the notions of smoothness and approximate smoothness can coincide. Example 2.7 (a) Observe that if x = (x , x , x ,...) ∈   {θ } then d(x ) = 2if 1 2 3 1 x = 0for some i ∈ N and d(x ) = 0if x = 0 for all i. Using this observation it i 0 i follows that x ∈   {θ } is ε-smooth for ε ∈[0, 2) if and only if x is smooth. (b) If x = (x , x , x ,...) ∈ c  {θ } then d(x ) = 2 if norm of x is attained at 1 2 3 0 more than one coordinate and d(x ) = 0 if norm of x is attained only at one coordinate. Thus, it follows that x ∈ c  {θ } is ε-smooth for ε ∈[0, 2) if and only if x is smooth. 2.2 Approximate rotundity Rotundity (strict convexity) is another important geometrical property of normed spaces. Although rotundity and smoothness are independent properties, they are related to each other. We would like to show that it is so with their approximate counterparts. Let X be a normed linear space and let ε ∈[0, 2). Definition 2.8 We say that X is ε-rotund (or ε-strictly convex) if S(X ) ≤ ε. Obviously, for ε = 0, the above condition means that each functional f ∈ S ∗ supports the unit sphere in at most one point, which is equivalent to rotundity (cf. [15, Theorem 5.1.15, Corollary 5.1.16]). Basedon(1.3) and (1.4), we can easily establish connections between approximate smoothness (rotundity) of a given space and approximate rotundity (smoothness) of its dual. Theorem 2.9 Let X be a normed linear space and let ε ∈[0, 2). 1. If X is ε-smooth, then X is ε-rotund; 2. If X is ε-rotund, then X is ε-smooth; 3. If X is reflexive, then X is ε-smooth if and only if X is ε-rotund and X is ε-rotund if and only if X is ε-smooth. 3 Approximate additivity of the Birkhoff–James orthogonality in approximately smooth spaces As we have reminded, the Birkhoff–James orthogonality is right-additive at smooth points. Although it is no longer true for non-smooth points, we will show that under a 41 Page 10 of 23 J. Chmieliński et al. Fig. 1 Illustration to Example 3.1 relaxed condition of approximate smoothness, right-additivity can be partially main- tained. We will start with an example showing that, in general, approximate smoothness does not imply approximate right-additivity of the Birkhoff–James orthogonality. Example 3.1 Let X = R with a norm generated by B := conv {(1, 1), (0, 1 + δ), (−1, 1), (−1, −1), (0, −1 − δ), (1, −1)} as a unit ball (with δ> 0). Consider the point P = (0, 1 + δ) and functionals δ 1 f (x , y) = x + y, 1+δ 1+δ δ 1 g(x , y) =− x + y. 1+δ 1+δ Notice that f , g ∈ J (P) (we denote by L and L the respective supporting lines) f g and J (P) = conv { f , g} (see Fig. 1). Then, 2δ 2δ diam J (P) = f − g= sup |x|= . 1 + δ 1 + δ (x ,y)∈B Thus, the considered space is ε-smooth at P, with arbitrarily small ε (if only δ is small enough). On the other hand, for R = (1,δ) and R = (−1,δ),wehave 1 2 P⊥ R , P⊥ R B 1 B 2 2δ ε but R + R = (0, 2δ) = λP with λ = hence P⊥  (R + R ) for any ε ∈[0, 1). 1 2 1 2 1+δ B Although approximate smoothness generally does not imply even approximate right-additivity (no matter how small is ε), the following results give some information in particular situations. Approximate smoothness in normed linear spaces Page 11 of 23 41 Theorem 3.2 Let X be a normed linear space and let x ∈ X {θ }. Let y , y ∈ X {θ } 1 2 ε ε be such that x ⊥ y ,x ⊥ y , where ε ∈[0, 1) is such that 1 2 B B 2y + y 1 2 0 ≤ ε< < 1. 3(y +y ) 1 2 If x is ε-smooth, then there exists ε ∈[0, 1) such that x ⊥ (y + y ). 1 1 2 Proof According to (1.8), we can find f , g ∈ J (x ) such that | f (y )|≤ εy  and 1 1 |g(y )|≤ εy .Now,using ε-smoothness of x, we get 2 2 | f (y )|=|( f − g)(y ) + g(y )| 2 2 2 ≤ f − gy +g(y ) 2 2 ≤ εy + εy 2 2 = 2εy . Using similar arguments, we can show that |g(y )|≤ 2εy . Convexity of J (x ) 1 1 1 1 yields f + g ∈ J (x ) and 2 2 1 1 1 f + g (y + y ) ≤ (εy + 2εy + 2εy + εy ) 1 2 1 2 1 2 2 2 2 = ε(y +y ) 1 2 < y + y . 1 2 1 1 Let ε ∈[0, 1) be such that |( f + g)(y + y )|≤ ε y + y . Then, (1.8) implies 1 1 2 1 1 2 2 2 that x ⊥ (y + y ). 1 2 Theorem 3.3 Let X be a normed linear space and let x ∈ X {θ }. Let y , y ∈ X {θ } 1 2 2y +y 1 2 be such that x ⊥ y ,x ⊥ y .If x is ε-smooth, where 0 ≤ ε< ≤ 2, then B 1 B 2 y +y 1 2 there exists ε ∈[0, 1) such that x ⊥ (y + y ). 1 1 2 Proof We find f , g ∈ J (x ) such that f (y ) = 0 and g(y ) = 0. Now, by ε-smoothness 1 2 of x, we get | f (y )|=|( f − g)(y )|≤ εy , 2 2 2 1 1 and similarly |g(y )|≤ εy . Clearly, f + g ∈ J (x ) and 1 1 2 2 1 1 ε f + g (y + y ) ≤ (y +y )< y + y . 1 2 1 2 1 2 2 2 2 1 1 Let ε ∈[0, 1) be such that f + g (y + y ) ≤ ε y + y .Thus,(1.8) implies 1 1 2 1 1 2 2 2 that x ⊥ (y + y ). 1 2 B 41 Page 12 of 23 J. Chmieliński et al. Fig. 2 Illustration to Example 3.5 We now provide an example to show that in Theorems 3.2 and 3.3, approximate smoothness of x cannot be omitted. We will need the following lemma. Lemma 3.4 Let X be a normed linear space and let x ∈ S . Then, the approximate Birkhoff–James orthogonality is not right-additive at x if and only if there exist y , y ∈ 1 2 y +y 1 2 S ,f , f ∈ J (x ) such that y ∈ / M ,y ∈ / M and ∈ M for all f ∈ J (x ). X 1 2 1 f 2 f f 1 2 y +y 1 2 Proof The proof relies on characterizations (1.9) and (1.10). Let x ∈ S and suppose that the approximate Birkhoff–James orthogonality is not right-additive at x, i.e., there ε ε exist y , y ∈ S , ε ∈[0, 1) such that x ⊥ y , x ⊥ y and there does not exist 1 2 X 1 2 B B y +y 1 1 2 any ε ∈[0, 1) such that x ⊥ . Equivalently, we can write that (due to (1.9)) B y +y 1 2 there exist f , f ∈ J (x ) such that y ∈ / M , y ∈ / M as well as (by (1.10)) that 1 2 1 f 2 f 1 2 y +y 1 2 ∈ M for all f ∈ J (x ). y +y 1 2 If x ∈ X  {θ } is ε-smooth and y , y ∈ X satisfy the conditions stated in Theo- 1 2 y +y 1 2 rem 3.2, then there exists f ∈ J (x ) such that ∈ / M . The following example y +y 1 2 shows that if in Theorems 3.2 or 3.3 the assumption of x being ε-smooth is omitted, the results are not true. Example 3.5 Consider X = (R ,  ).Let x , y , y ∈ S , where x = (1, 1), y = ∞ 1 2 X 1 (1,α), y = (α, 1),0 <α < (see Fig. 2A). Then, we can observe, using (1.7), that 2y +y α α 1+α 1 2 x ⊥ y and x ⊥ y . Moreover, 0 <α < = . 1 2 B B 3 3(y +y ) 1 2 y +y 1 2 Now, we have = x ∈ M for all f ∈ J (x ) and, by Lemma 3.4, approximate y +y 1 2 Birkhoff–James orthogonality is not right-additive at x. To justify essentialness of approximate smoothness in Theorem 3.3 take x , z , z ∈ 1 2 S , where x = (1, 1), z = (1, −α), z = (−α, 1),0 <α < (see Fig. 2B). Then, X 1 2 z +z 1 2 x ⊥ z , x ⊥ z and clearly, = x ∈ M for all f ∈ J (x ).Thus,Lemma 3.4 B 1 B 2 f z +z 1 2 implies that Birkhoff–James orthogonality is not approximately right-additive at x. Finally, in this section, we give a result showing that approximate smoothness of a vector x guarantees the approximate right-additivity of the Birkhoff–James orthogo- nality on some restricted set of vectors. Approximate smoothness in normed linear spaces Page 13 of 23 41 Theorem 3.6 Let X be a real normed linear space and let x ∈ X  {θ } be ε-smooth y +y 1 2 with ε ∈[0, 2). Suppose that y , y ∈ X are such that min{y , y } ≤ . 1 2 1 2 If x ⊥ y and x ⊥ y , then B 1 B 2 ε/2 x ⊥ (y + y ). 1 2 Proof Since x ⊥ y and x ⊥ y , there exist f , g ∈ J (x ) such that f (y ) = g(y ) = B 1 B 2 1 2 0. Therefore, f (y ) = ( f − g)(y ) and g(y ) = (g − f )(y ) 2 2 1 1 which, together with the assumed diam J (x ) ≤ ε,gives | f (y )|≤ εy  and |g(y )|≤ εy . 2 2 1 1 y +y 1 2 Suppose that y ≤y . It follows then from the assumption that y ≤ 1 2 1 and |g(y + y )|=|g(y )|≤ εy ≤ y + y . 1 2 1 1 1 2 ε/2 This means that x ⊥ (y + y ). 1 2 Similarly, if y ≤y , we show that | f (y + y )|≤ y + y , which also 2 1 1 2 1 2 gives the assertion. y +y 1 2 Note that the condition min{y , y } ≤ depends both on directions 1 2 and norms of vectors y , y . It holds true, however, regardless of directions, if y ≥ 1 2 1 3y  or y ≥ 3y . 2 2 1 4 Polyhedral spaces Now, we consider a 2-dimensional regular polyhedral Banach space X with 2n extreme points. Regularity here means that all the edges of the unit sphere are of the same length with respect to the Euclidean metric and all the interior angles are of the same measure. For such spaces, we will calculate the values of d(x ) for each x ∈ Ext B and the value of E (X ). Example 4.1 Let X be a 2-dimensional regular polyhedral Banach space with 2n (n ≥ 2) extreme points and let x ∈ Ext B . Then, 2tan if n is even, 2n d(x ) = E (X ) = π (n−1)π 2tan sin if n is odd. 2n 2n Proof If x ∈ S Ext B then clearly d(x ) = 0. Thus, to calculate E (X ) it is sufficient X X to find d(x ) for all x ∈ Ext B . Moreover, regularity and symmetry of S implies that X X d(x ) = d(y) for all x , y ∈ Ext B . X 41 Page 14 of 23 J. Chmieliński et al. Fig. 3 Illustration to Example 4.1 Without loss of generality, we may assume that Ext B ={v : 1 ≤ k ≤ 2n}, X k (k−1)π (k−1)π where v = cos , sin for 1 ≤ k ≤ 2n (see Fig. 3). n n Let f be the unique support functional for the segment v v for k = 1,..., 2n−1 k k k+1 and v v for k = 2n (on Fig. 3 the respective supporting lines are denoted by L 2n 1 f and L ). Some calculations, which will be omitted here, lead to the explicit formula 2n for the value of f at (x , y) ∈ R : (2k − 1)π (2k − 1)π π f ((x , y)) = x cos + y sin sec 2n 2n 2n (compare with a similar result in [19, the proof of Theorem 3.1]). Moreover, f f if 1 < k ≤ 2n, k−1 k J (v ) = f f if k = 1 2n 1 and thus f − f  if 1 < k ≤ 2n, k−1 k d(v ) = f − f  if k = 1. 2n 1 Approximate smoothness in normed linear spaces Page 15 of 23 41 For 1 < k ≤ 2n,wehave (2k − 1)π (2k − 3)π π ( f − f )((x , y)) = x cos − cos sec k k−1 2n 2n 2n (2k − 1)π (2k − 3)π π + y sin − sin sec 2n 2n 2n (k − 1)π π (k − 1)π π π = −2x sin sin + 2y cos sin sec . n 2n n 2n 2n To calculate the norm of f − f (supremum over the unit sphere), we use the k k−1 Krein–Milman theorem and restrict ourselves to extremal points. Thus, we have for 1 < k ≤ 2n f − f = k k−1 (4.1) (k−1)π π (k−1)π π π max −2x sin sin + 2y cos sin sec . (x ,y)∈Ext B n 2n n 2n 2n If n is even then by taking k = + 1in(4.1), we get π π f n − f n = max 2|x | sin sec . 2 2 (x ,y)∈Ext B 2n 2n n n For (x , y) ∈ Ext B , |x|≤ 1. Thus,  f − f = 2tan . X +1 2n 2 2 If n is odd, then by taking k = n + 1in (4.1), we get π π f − f = max 2|y| sin sec . (4.2) n+1 n (x ,y)∈Ext B 2n 2n (n−1)π For (x , y) ∈ Ext B , |y|≤ sin . Thus, 2n (n − 1)π π π π (n − 1)π f − f = 2sin sin sec = 2tan sin . (4.3) n+1 n 2n 2n 2n 2n 2n Now, using (4.2), (4.3), symmetry and regularity of S , we get 2tan if n is even, 2n E (X ) = d(v ) = π (n−1)π 2tan sin if n is odd. 2n 2n As an application of the above, we obtain the following result on approximate smoothness of a 2-dimensional regular polyhedral Banach space X with 2n extreme points. Proposition 4.2 Let X be a 2-dimensional regular polyhedral Banach space with 2n extreme points (n ≥ 2). Then, the following holds true. (i) X is ε-smooth for ε ∈[2tan , 2), if n is even. 2n 41 Page 16 of 23 J. Chmieliński et al. π (n−1)π (ii) X is ε-smooth for ε ∈[2tan sin , 2),ifn ≥ 3 is odd. 2n 2n In the next result, we give formulas for d(x ), x ∈ S and E (X ) for a finite-dimensional polyhedral Banach space X. Note that in case of X being a finite- dimensional polyhedral Banach space, its dual X is also a polyhedral Banach space with finitely many extreme points in B . Theorem 4.3 Let X be a finite-dimensional polyhedral Banach space and let x ∈ S . Then, (i) d(x ) = max{ f − f : f , f ∈ Ext B such that f , f ∈ J (x )}, i j i j X i j + + (ii) E (X ) = max{ f − f : f , f ∈ Ext B such that M ∩ M =∅}. i j i j X f f i j Proof (i) Let x ∈ S and F , F ,..., F be the facets of S which contain X i i i X 1 2 k x.Let f , f ,..., f ∈ Ext B be the unique supporting functionals for the i i i X 1 2 facets F , F ,..., F , respectively. Let f , g ∈ J (x ). Then, f = α f , i i i j i 1 2 k j =1 j k k k g = β f , where 0 ≤ α ,β ≤ 1 and α = 1, β = 1. Now, j i j j j j j =1 j j =1 j =1 k k k f − g= α f − g = α f − α g j i j i j j j j =1 j =1 j =1 k k = α ( f − g) ≤ α  f − g. j i j i j j j =1 j =1 Similar arguments show that f − g≤ β  f − f ≤ max  f − f i  i i i i j j  j 1≤≤k =1 for all 1 ≤ j ≤ k. Thus,  f − g≤ max  f − f  and this proves the result. i i 1≤ j ,≤k (ii) Observe that if y ∈ F for some facet F and f , g ∈ J (y) then f , g ∈ J (x ) for some x ∈ Ext B ∩ F. Thus, to calculate E (X ) it is sufficient to consider x ∈ Ext B . X X Now, (ii) follows from (i) by observing that if f , g ∈ J (x ) for some x ∈ Ext B ∩ F and facet F then M ∩ M =∅. 5 Direct sums Given normed linear spaces X , Y , we study the space Z = X ⊕ Y,1 ≤ p ≤∞, and approximate smoothness of its elements. We start with a description of the set of supporting functionals. Proposition 5.1 Let X , Y be normed linear spaces and let Z = X ⊕ Y with 1 ≤ p ≤ ∞. Approximate smoothness in normed linear spaces Page 17 of 23 41 1. If 1 < p < ∞ and q is conjugated to p, then for any x ∈ X  {θ } and y ∈ Y  {θ }, we have (a) J ((x , y)) = p−1 p−1 x  f y g ∗ ∗ , ∈ S : f ∈ J (x ), g ∈ J (y) , X ⊕ Y 1 1 q p p q p p q (x  +y ) (x  +y ) (5.1) ∗ ∗ (b) J ((x,θ)) ={( f ,θ) ∈ S : f ∈ J (x )}, X ⊕ Y ∗ ∗ (c) J ((θ , y)) ={(θ , g) ∈ S : g ∈ J (y)}. X ⊕ Y 2 If p = 1,x ∈ X  {θ },y ∈ Y  {θ }, then (a) J ((x , y)) ={( f , g) ∈ S ∗ ∗ : f ∈ J (x ), g ∈ J (y)}, X ⊕ Y (b) J ((x,θ)) ={( f , g) ∈ S ∗ ∗ : f ∈ J (x ), g ∈ B ∗ }, X ⊕ Y Y (c) J ((θ , y)) ={( f , g) ∈ S ∗ ∗ : f ∈ B ∗ , g ∈ J (y)}. X ⊕ Y X 3 Let p =∞ and (x , y) ∈ Z  {θ }. (a) If x  > y, then J ((x , y)) ={( f ,θ) ∈ S ∗ ∗ : f ∈ J (x )}. X ⊕ Y (b) If x  < y, then ∗ ∗ J ((x , y)) ={(θ , g) ∈ S : g ∈ J (y)}. X ⊕ Y (c) If x=y, then {(α f ,(1 − α)g) : f ∈ J (x ), g ∈ J (y), 0 ≤ α ≤ 1}⊆ J ((x , y)). Proof (1a).Let F ∈ J ((x , y)), where F = ( f , g) ∈ S ∗ ∗. Then, using Hölder’s X ⊕ Y inequality, we get p p (x  +y ) =(x , y) = F ((x , y)) = f (x ) + g(y) ≤ f x+gy 1 1 q q p p q p ≤ ( f  +g ) (x  +y ) p p = (x  +y ) . This shows that equality holds in Hölder’s inequality and thus p−1 p−1 x  y f = , g= . (5.2) 1 1 p p p p q q (x  +y ) (x  +y ) 41 Page 18 of 23 J. Chmieliński et al. Also, we get f (x ) = f x , g(y) =gy which implies that f = ∈ J (x ) and g ˜ = ∈ J (y). Combining this with (5.2), we get p−1 p−1 x  f y g ˜ F = ( f , g) = , 1 1 p p p p q q (x  +y ) (x  +y ) which shows ⊆ in (5.1). Now, consider p−1 p−1 x  f y g ∗ ∗ F = , ∈ S , X ⊕ Y 1 1 p p p p q q (x  +y ) (x  +y ) where f ∈ J (x ) and g ∈ J (y). Then, (we use that (p − 1)q = p) p p x  +y F ((x , y)) = =(x , y) . p p (x  +y ) Thus, F ∈ J ((x , y)) and this proves ⊇ in (5.1) and finishes the proof of part (a). Parts (b) and (c) follow using the similar reasoning. (2). For the proof of (a) let x ∈ X \{θ }, y ∈ Y \{θ }, f ∈ J (x ) and g ∈ J (y). Then, ∗ ∗ ( f , g) ∈ S and ( f , g)((x , y)) = f (x ) + g(y) =x+y=(x , y) . X ⊕ Y 1 Thus, ( f , g) ∈ J ((x , y)). For the reverse, let ( f , g) ∈ J ((x , y)). Then,  f ≤ 1, g≤ 1 and x+y=(x , y) = ( f , g)((x , y)) = f (x ) + g(y) ≤ f x+gy≤x+y. This shows that  f = 1, g= 1, f (x ) =x  and g(y) =y. Thus, f ∈ J (x ) and g ∈ J (y). ∗ ∗ ∗ To prove (b) let x ∈ X \{θ }, f ∈ J (x ) and g ∈ B . Then, ( f , g) ∈ S and Y X ⊕ Y ( f , g)((x,θ)) = f (x ) =x=(x,θ) . Thus, ( f , g) ∈ J ((x,θ)). The reverse inclusion is clear. The proof of (c) is analogous. (3). For the proof of ((a))let x ∈ X \{θ }, y ∈ Y be such that x  > y.Let f ∈ ∗ ∗ J (x ). Then, clearly ( f ,θ) ∈ S , ( f , θ )((x , y)) = f (x ) =x=(x , y) X ⊕ Y ∞ and thus ( f ,θ) ∈ J ((x , y)). Now, let ( f , g) ∈ J ((x , y)), where g ∈ B .If f = θ, then ( f , g)((x , y)) = g(y) ≤y < (x , y) . Thus, f = θ. Suppose g = θ. Since ( f , g) ∈ S ∗ ∗, f = θ, g = θ,gives  f  < 1 X ⊕ Y and g < 1. Also,  f +g= 1 implies g= 1 − f . Thus, x=(x , y) = ( f , g)((x , y)) = f (x ) + g(y) ≤ f x+gy <  f x+ (1 − f )x=x . Approximate smoothness in normed linear spaces Page 19 of 23 41 The obtained contradiction proves the required form of J ((x , y)). The proof of (b) is analogous. Finally, we prove (c).Let x ∈ X \{θ }, y ∈ Y \{θ } be such that x=y.Let ∗ ∗ f ∈ J (x ), g ∈ J (y) and 0 ≤ α ≤ 1. Then, (α f ,(1 − α)g) ∈ S and (α f ,(1 − X ⊕ Y α)g)((x , y)) = α f (x ) + (1 − α)g(y) = αx+ (1 − α)y=x=(x , y) . This proves the result. Corollary 5.2 Let X , Y be normed linear spaces and let Z = X ⊕ Y with 1 ≤ p ≤∞. 1. If 1 < p < ∞ and q is conjugated to p, then for any x ∈ X  {θ } and y ∈ Y  {θ }, we have (a) p p x  y q q d((x , y)) = d(x ) + d(y) , (5.3) p p p p x  +y x  +y (b) d((x,θ)) = d(x ), (c) d((θ , y)) = d(y). 2. If p = 1,x ∈ X  {θ },y ∈ Y  {θ }, then (a) d((x , y)) = max{d(x ), d(y)}, (b) d((x,θ)) = 2, (c) d((θ , y)) = 2. 3. Let p =∞, (x , y) ∈ Z  {θ }. (a) If x  > y, then d((x , y)) = d(x ). (b) If x  < y, then d((x , y)) = d(y). (c) If x=y, then d((x , y)) = 2. Proof We will prove (1a).Let F , G ∈ J ((x , y)). On account of (5.1), there exist f , g ∈ J (x ), f , g ∈ J (y) such that x x y y p−1 p−1 x  f y f x y F = , 1 1 p p p p q q (x  +y ) (x  +y ) and p−1 p−1 y g x  g x y G = , . 1 1 p p q p p q (x  +y ) (x  +y ) This gives (using (p − 1)q = p) p p x  y q q F − G =  f − g  +  f − g q x x y y p p p p x  +y x  +y 41 Page 20 of 23 J. Chmieliński et al. and thus, p p x  y q q q d((x , y)) ≤ d(x ) + d(y) . p p p p x  +y x  +y δ δ To prove the reverse inequality fix arbitrarily δ> 0 and choose f , g ∈ J (x ) such x x δ δ q q δ δ that  f − g  > d(x ) − δ. Analogously, let f , g ∈ J (y) be chosen such that x x y y δ δ q q f − g  > d(y) − δ. Define y y p−1 δ p−1 δ y f x  f δ x F := , 1 1 p p q p p q (x  +y ) (x  +y ) and p−1 δ p−1 δ y g x  g G := , . 1 1 p p p p q q (x  +y ) (x  +y ) δ δ By (5.1), F , G ∈ J ((x , y)) whence q δ δ d((x , y)) ≥F − G p p x  y q q > (d(x ) − δ) + (d(y) − δ) p p p p x  +y x  +y and since δ> 0 was arbitrary, we get p p x  y q q q d((x , y)) ≥ d(x ) + d(y) . p p p p x  +y x  +y The proofs of other cases are similar or obvious. Now, as a straightforward consequence of Corollary 5.2, we characterize approxi- mate smoothness of the direct sum. Theorem 5.3 Let X, Y be normed linear spaces and Z = X ⊕ Y, 1 < p < ∞. Then, the following statements hold true: (i) Let x ∈ X  {θ },y ∈ Y  {θ }. (a) If x is ε -smooth in X and y is ε -smoothinY for ε ,ε ∈[0, 2), then (x , y) x y x y is ε-smooth in Z with p p x  y q q ε := ε + ε . x y p p p p x  +y x  +y (b) If x is ε-smooth in X and y is ε-smoothinY for ε ∈[0, 2), then (x , y) is ε-smooth in Z. Approximate smoothness in normed linear spaces Page 21 of 23 41 (c) If (x , y) is ε-smooth in Z for ε ∈[0, 2), then either x is ε-smooth in X or y is ε-smoothinY. (d) (x , y) is approximately smooth if and only if either x or y is approximately smooth. (ii) Let x ∈ X  {θ }. Then, (x,θ) ∈ Zis ε-smooth for ε ∈[0, 2) if and only if x is ε-smooth in X. (iii) Let y ∈ Y  {θ }. Then, (θ , y) ∈ Zis ε-smooth for ε ∈[0, 2) if and only if y is ε-smoothinY. Proof For the proof of (i), we need to apply Corollary 5.2 (1a). In particular, for (id), we observe that d(x , y)< 2 if and only if d(x)< 2or d(y)< 2. The statements (ii) and (iii) immediately follow from Corollary 5.2 (1b) and (1c), respectively. Theorem 5.4 Let X, Y be normed linear spaces and Z = X ⊕ Y . Then, the following hold true: (i) (a) If x is ε -smooth in X and y is ε -smoothinY for ε ,ε ∈[0, 2), then (x , y) x y x y is ε-smooth in Z with ε := max{ε ,ε }. x y (b) If x is ε-smooth in X and y is ε-smoothinY for ε ∈[0, 2), then (x , y) is ε-smooth in Z. (c) If (x , y) is ε-smooth in Z for ε ∈[0, 2), then both x and y are ε-smooth in X and Y , respectively. (d) (x , y) is approximately smooth if and only if both x and y are approximately smooth. (ii) If x ∈ X  {θ } then (x,θ) ∈ Z cannot be approximately smooth. (iii) If y ∈ Y  {θ } then (θ , y) ∈ Z cannot be approximately smooth. Proof The proof of (i) relies on Corollary 5.2 (2a). Statements (ii) and (iii) follow immediately from Corollary 5.2 (2b) and (2c), respectively. Theorem 5.5 Let X, Y be normed linear spaces, Z = X ⊕ Y and z = (x , y) ∈ Z  {θ }. Then, the following statements hold true: (i) Let x  > y. Then, z is ε-smooth in Z for ε ∈[0, 2) if and only if x is ε-smooth in X. (ii) Let x  < y. Then, z is ε-smooth in Z for ε ∈[0, 2) if and only if y is ε-smooth in Y . (iii) If x=y, then z cannot be approximately smooth. Proof Statements (i) and (ii) follow from Corollary 5.2 (3a) and (3b), whereas (iii) is a consequence of Corollary 5.2 (3c). The following final example is an application of the above theorem and the fact that any t ∈ R {0} is a smooth point of R. Example 5.6 Let Z = X ⊕ R be a 3-dimensional polyhedral Banach space whose unit ball is a right prism with regular polyhedron P as its base. Then, the following hold true for z = (x , t ) ∈ Z  {θ }: 41 Page 22 of 23 J. Chmieliński et al. (i) Let x  > |t |. Then, z is ε-smooth in Z for ε ∈[0, 2) if and only if x is ε-smooth in X. (ii) Let x  < |t |. Then, z is smooth. (iii) If x=|t |, then z cannot be approximately smooth. Acknowledgements The research of Divya Khurana is sponsored by Dr. D. S. Kothari Postdoctoral Fel- lowship under the mentorship of Professor Gadadhar Misra. She would also like to thank Professor Gideon Schechtman for discussion on the direct sum of normed linear spaces. The research of Dr. Debmalya Sain is sponsored by DST-SERB under the mentorship of Professor Apoorva Khare. Dr. Sain is elated to acknowl- edge the monumental positive role played by the Department of Mathematics, Indian Institute of Science, for providing him with a rich mathematical setting. The authors thank the referees for their valuable comments. Declarations Conflict of interest The authors state no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Alsina, C., Sikorska, J., Santos Tomás, M.: Norm Derivatives and Characterizations of Inner Product Spaces. World Scientific, Hackensack (2010) 2. Birkhoff, G.: Orthogonality in linear metric spaces. Duke Math. J. 1, 169–172 (1935) 3. Chmielinski, ´ J.: On an -Birkhoff orthogonality. J. Inequal. Pure Appl. Math. 6(3), Article 79 (2005) 4. Chmielinski, ´ J.: Approximate Birkhoff–James orthogonality in normed linear spaces and related top- ics. In: Aron, R.M., Moslehian, M.S., Spitkovsky, I.M., Woerdeman, H.J. (eds.) Operator and Norm Inequalities and Related Topics, 303–320. Birkhäuser, Cham (2022) 5. Chmielinski, ´ J., Khurana, D., Sain, D.: Local approximate symmetry of Birkhoff–James orthogonality in normed linear spaces. Results Math. 76, 136 (2021) 6. Chmielinski, ´ J., Stypuła, T., Wójcik, P.: Approximate orthogonality in normed spaces and its applica- tions. Linear Algebra Appl. 531, 305–317 (2017) 7. Day, M.M.: Normed Linear Spaces. Springer, Berlin (1973) 8. Van Dulst, D.: Reflexive and Superreflexive Banach Spaces, Mathematical Centre Tracts 102. Mathe- matisch Centrum, Amsterdam (1978) 9. Dragomir, S.S.: Semi-Inner Products and Applications. Nova Science Publishers, Hauppauge, NY (2004) 10. Fabian, M., Montesinos, V., Zizler, V.: Smoothness in Banach spaces. Selected problems. Rev. R. Acad. Cienc Exactas Fís. Nat. Ser. A Mat. 100, 101–125 (2006) 11. James, R.C.: Orthogonality and linear functionals in normed linear spaces. Trans. Am. Math. Soc. 61, 265–292 (1947) 12. James, R.C.: Inner product in normed linear spaces. Bull. Am. Math. Soc. 53, 559–566 (1947) 13. Johnson, W.B., Lindenstrauss, J.: Basic concepts in the geometry of Banach spaces. In: Johnson, W.B, Lindenstrauss J. (eds), Handbook of geometry of Banach spaces, vol. 1, North-Holland Publishing Co., Amsterdam, pp. 1–84 (2001) 14. Mal, A., Paul, K., Rao, T.S.S.R.K., Sain, D.: Approximate Birkhoff–James orthogonality and smooth- ness in the space of bounded linear operators. Monatsh. Math. 190(3), 549–558 (2019) Approximate smoothness in normed linear spaces Page 23 of 23 41 15. Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer-Verlag, New York (1998) 16. Paul, K., Sain, D., Mal, A.: Approximate Birkhoff–James orthogonality in the space of bounded linear operators. Linear Algebra Appl. 537, 348–357 (2018) 17. Sain, D., Paul, K., Bhunia, P., Bag, S.: On the numerical index of polyhedral Banach spaces. Linear Algebra Appl. 577, 121–133 (2019) 18. Sain, D., Paul, K., Mal, A.: On approximate Birkhoff–James orthogonality and normal cones in a normed space. J. Convex Anal. 26(1), 341–351 (2019) 19. Sain, D., Roy, S., Bagchi, S., Balestro, V.: A study of symmetric points in Banach spaces. Linear Multilinear Algebra 70(5), 888–898 (2020) 20. Stypuła, T., Wójcik, P.: Characterizations of rotundity and smoothness by approximate orthogonalities. Ann. Math. Sil. 30, 183–201 (2016)

Journal

Banach Journal of Mathematical Analysis – Springer Journals

Published: Jul 1, 2023

Keywords: Smoothness; Rotundity; Approximate smoothness; Approximate rotundity; Birkhoff–James orthogonality; Approximate Birkhoff–James orthogonality; Supporting hyperplanes; Polyhedral spaces; Direct sums; 46B20; 47L05; 51F20; 52B11

There are no references for this article.