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Notes on Functional Analysis

Part of the Texts and Readings in Mathematics Book Series (volume 50)

DOI: 10.1007/978-93-86279-45-3

ISBN: 978-81-85931-89-0

Publisher: Hindustan Book Agency - Springer Journals

May 24, 2017

1 - 10 of 26 Chapters

LitStream

Chapter 1Banach Spaces

Bhatia, Rajendra

Pages: 1 –10

[The subject Functional Analysis was created at the beginning of the twentieth century to provide a unified framework for the study of problems that involve continuity and linearity. The basic objects of study in this subject are Banach spaces and linear operators on these spaces.]
LitStream

Chapter 2Dimensionality

Bhatia, Rajendra

Pages: 11 –18

[Let X be a vector space and let S be a subset of it. We say S is linearly independent if for every finite subset {x1,…, xn} of S, the equation (2.1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}...
LitStream

Chapter 3New Banach Spaces from Old

Bhatia, Rajendra

Pages: 19 –27

[Let X be a vector space and M a subspace of it. Say that two elements x and y of X are equivalent, x ~ y, if x − y ∈ M. This is an equivalence relation on X. The coset of x under this relation is the set \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}...
LitStream

Chapter 4The Hahn-Banach Theorem

Bhatia, Rajendra

Pages: 28 –35

[The Hahn-Banach Theorem (H.B.T.) is called one of the three basic principles of linear analysis—the two others are the Uniform Boundedness Principle and the Open Mapping Theorem. We will study them in the next three lectures. The H.B.T. has several versions and several corollaries.]
LitStream

Chapter 5The Uniform Boundedness Principle

Bhatia, Rajendra

Pages: 36 –41

[The Baire Category Theorem says that a complete metric space cannot be the union of a countable number of nowhere dense sets. This has several very useful consequences. One of them is the Uniform Boundedness Principle (U.B.P.) also called the Banach-Steinhaus Theorem.]
LitStream

Chapter 6The Open Mapping Theorem

Bhatia, Rajendra

Pages: 42 –48

[Theorems that tell us that a continuous map is also open under some simple conditions play a very important role in analysis. The open mapping theorem is one such result.]
LitStream

Chapter 7Dual Spaces

Bhatia, Rajendra

Pages: 49 –57

[The idea of duality, and the associated notion of adjointness, are important in functional analysis. We will identify the spaces X* for some of the standard Banach spaces.]
LitStream

Chapter 8Some Applications

Bhatia, Rajendra

Pages: 58 –65

[Let μn be a sequence of probability measures on [0, 1]. Then there exists a subsequence μm and a probability measure μ such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy}...
LitStream

Chapter 9The Weak Topology

Bhatia, Rajendra

Pages: 66 –72

[When we say that a sequence fn in the space C[0, 1] converges to f, we mean that ‖fn − f‖ → 0 as n → ∞; and this is the same as saying fn converges to f uniformly. There are other notions of convergence that are weaker, and still very useful in analysis. This is the motivation for studying...
LitStream

Chapter 10The Second Dual and the Weak* Topology

Bhatia, Rajendra

Pages: 73 –80

[The dual of X* is another Banach space X**. This is called the second dual or the bidual of X. Let J be the map from X into X** that associates with x ∈ X the element Fx ∈ X** defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts}...

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