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2025 | Book

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Introduction to Functional Analysis

Authors: Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira

Publisher: Springer Nature Switzerland

Book Series : Universitext

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

Table of Contents
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About this book

This textbook offers an accessible introduction to Functional Analysis, providing a solid foundation for students new to the field. It is designed to support learners with no prior background in the subject and serves as an effective guide for introductory courses, suitable for students in mathematics and other STEM disciplines.

The book provides a comprehensive introduction to the essential topics of Functional Analysis across the first seven chapters, with a particular emphasis on normed vector spaces, Banach spaces, and continuous linear operators. It examines the parallels and distinctions between Functional Analysis and Linear Algebra, highlighting the crucial role of continuity in infinite-dimensional spaces and its implications for complex mathematical problems.

Later chapters broaden the scope, including advanced topics such as topological vector spaces, techniques in Nonlinear Analysis, and key theorems in theory of Banach spaces. Exercises throughout the book reinforce understanding and allow readers to test their grasp of the material.

Designed for students in mathematics and other STEM disciplines, as well as researchers seeking a thorough introduction to Functional Analysis, this book takes a clear and accessible approach. Prerequisites include a strong foundation in analysis in the real line, linear algebra, and basic topology, with helpful references provided for additional consultation.

Table of Contents

Frontmatter
Chapter 1. Normed Vector Spaces
Abstract
In vector spaces, we know how to add their elements and multiply their elements by scalars, while in metric spaces we know how to calculate the distance between two of their elements. A normed space is the mathematical structure in which all this can be done in the same environment.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 2. Continuous Linear Operators
Abstract
Normed spaces have an algebraic structure of a vector space to which linear transformations are associated, and a topological structure of a metric space to which continuous functions are associated. Thus, the morphisms between normed spaces are functions that are simultaneously linear and continuous. Such functions are usually called continuous linear operators.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 3. Hahn–Banach Theorems
Abstract
It is difficult to measure the relevance of the Hahn–Banach Theorem in Functional Analysis, given the numerous corollaries and applications it has. It also finds applications in other areas of mathematics; for example, Complex Analysis, Measure Theory, Control Theory, Convex Programming, and Game Theory.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 4. Duality and Reflexive Spaces
Abstract
We will begin this chapter by showing that it is possible to describe all continuous linear functionals on some of the classical spaces we have been working with. The relevant feature of this description of dual spaces is that the functionals are described as objects belonging to the same class as the elements of the space, namely, continuous linear functionals on function spaces shall be described as functions, and continuous linear functionals on sequence spaces shall be described as sequences.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 5. Hilbert Spaces
Abstract
From Calculus and Analytical Geometry courses, the reader knows that the usual inner product of \(\mathbb {R}^n\)
$$\displaystyle \langle \cdot , \cdot \rangle \colon \mathbb {R}^{n}\times \mathbb {R}^{n}\longrightarrow \mathbb {R}~~,~(x,y) \mapsto \langle x,y \rangle := \sum _{j=1}^nx_j y_j, $$
where \(x = (x_1, \ldots , x_n)\) and \(y = (y_1, \ldots , y_n)\), is an essential tool in the construction of the theory.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 6. Weak Topologies
Abstract
In Theorem 1.​5.​4 we proved that the closed unit ball of an infinite-dimensional normed space is not compact. This strong contrast with finite dimension indicates that the existence of compact sets that are not contained in finite-dimensional subspaces is a delicate issue. Examples exist: we saw in Exercise 1.​8.​35 that the Hilbert cube is compact in \(\ell _2\). The issue is not just the existence or non-existence of many compact sets; the central point is that those sets that we would like to be compact, namely the closed balls, are not.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 7. Spectral Theory of Compact Self-adjoint Operators
Abstract
Just as in Linear Algebra, in Functional Analysis the study of eigenvalues and eigenvectors of a linear and continuous operator is very useful.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 8. Topological Vector Spaces
Abstract
Let A be a subset of a normed space E. We ask the reader for a bit of patience to follow us in the proof of the following implication:
$$\displaystyle {} \mathrm {If}~ A ~\mathrm {is}~\mathrm {convex},~\mathrm {then}~ \overline {A} ~\mathrm {is}~\mathrm {also}~\mathrm {convex}. $$
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 9. Introduction to Nonlinear Analysis
Abstract
“…the world is nonlinear”. In this chapter, we distance ourselves a bit from the rigid, yet very useful linear structure of the previous chapters, to venture into a nonlinear world. The aim of this chapter is to offer a small sample of the theory of nonlinear operators between Banach spaces, introducing some basic tools in the study of nonlinear problems, usually found in mathematical models.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Chapter 10. Elements of Banach Space Theory
Abstract
Since the early days of Functional Analysis during the first decades of the twentieth century, the theory of Banach spaces has undergone quick and deep development, progressing both horizontally and vertically. The term “horizontal” is used to denote the expansion of the theory’s boundaries and the establishment of connections with other areas. Illustrations of this include Chap. 9 on Nonlinear Analysis, the development of Distribution Theory and its interconnections, the application of Sobolev spaces to the theory of Partial Differential Equations, the exploration of Banach Algebras (specifically C*-algebras), and the applications of the Hahn–Banach Theorem outlined at the beginning of Chap. 3.
Geraldo Botelho, Daniel Pellegrino, Eduardo Teixeira
Backmatter
Metadata
Title
Introduction to Functional Analysis
Authors
Geraldo Botelho
Daniel Pellegrino
Eduardo Teixeira
Copyright Year
2025
Electronic ISBN
978-3-031-81791-5
Print ISBN
978-3-031-81790-8
DOI
https://doi.org/10.1007/978-3-031-81791-5

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