# Functional analysis: Wikis

Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.

### Did you know ...

More interesting facts on Functional analysis

# Encyclopedia

For functional analysis as used in psychology, see the functional analysis (psychology) article.

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. This usage of the word functional goes back to the calculus of variations, implying a function whose argument is a function. Its use in general has been attributed to Italian mathematician and physicist Vito Volterra and its founding is largely attributed to a group of Polish mathematicians around Stefan Banach. In the modern view, functional analysis is seen as the study of vector spaces endowed with a topology, in particular infinitely dimensional spaces. In contrast, linear algebra deals mostly with finite dimensional spaces, or does not use topology. An important part of functional analysis is the extension of the theory of measure, integration, and probability to infinite dimensional spaces, also known as infinite dimensional analysis.

## Normed vector spaces

The basic and historically first class of spaces studied in functional analysis are complete normed vector spaces over the real or complex numbers. Such spaces are called Banach spaces. An important example is a Hilbert space, where the norm arises from an inner product. These spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics.

More generally, functional analysis includes the study of Fréchet spaces and other topological vector spaces not endowed with a norm.

An important object of study in functional analysis are the continuous linear operators defined on Banach and Hilbert spaces. These lead naturally to the definition of C*-algebras and other operator algebras.

### Hilbert spaces

Hilbert spaces can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. Since finite-dimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Aleph-null (ℵ0) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Aleph-null, and its morphisms. One of the open problems in functional analysis is to prove that every bounded linear operator on a Hilbert space has a proper invariant subspace. Many special cases of this invariant subspace problem have already been proven.

### Banach spaces

General Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example.

Examples of Banach spaces are Lp-spaces, for any real number $p\geq1$ (see Lp spaces). Given $p\geq1$ and a set X (which may be countable or uncountable), Lp(X) consists of "all Lebesgue-measurable functions whose absolute value's p-th power has finite integral".

That is, it consists of all Lebesgue-measurable functions f for which $\int_{x\in X}\left|f(x)\right|^p\,dx<+\infty$.

If X is countable, the integral may be replaced with a sum: $\sum_{X}\left|f(x)\right|^p<+\infty$, although for countable X, the space is usually denoted lp(X).

In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear functionals. The dual of the dual is not always isomorphic to the original space, but there is always a natural monomorphism from a space into its dual's dual. This is explained in the dual space article.

Also, the notion of derivative can be extended to arbitrary functions between Banach spaces. See, for instance, the Fréchet derivative article.

## Major and foundational results

Important results of functional analysis include:

## Foundations of mathematics considerations

Most spaces considered in functional analysis have infinite dimension. To show the existence of a vector space basis for such spaces may require Zorn's lemma. However, a somewhat different concept, Schauder basis, is usually more relevant in functional analysis. Many very important theorems require the Hahn-Banach theorem, usually proved using axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices. The Baire category theorem, needed to prove many important theorems, also requires a form of axiom of choice.

## Points of view

Functional analysis in its present form includes the following tendencies:

## References

• Brezis, H.: Analyse Fonctionnelle, Dunod ISBN 978-2100043149 or ISBN 978-2100493364
• Conway, John B.: A Course in Functional Analysis, 2nd edition, Springer-Verlag, 1994, ISBN 0-387-97245-5
• Dunford, N. and Schwartz, J.T. : Linear Operators, General Theory, and other 3 volumes, includes visualization charts
• Eidelman, Yuli, Vitali Milman, and Antonis Tsolomitis: Functional Analysis: An Introduction, American Mathematical Society, 2004.
• Giles,J.R.: Introduction to the Analysis of Normed Linear Spaces,Cambridge University Press,2000
• Hirsch F., Lacombe G. - "Elements of Functional Analysis", Springer 1999.
• Hutson, V., Pym, J.S., Cloud M.J.: Applications of Functional Analysis and Operator Theory, 2nd edition, Elsevier Science, 2005, ISBN 0-444-51790-1
• Kolmogorov, A.N and Fomin, S.V.: Elements of the Theory of Functions and Functional Analysis, Dover Publications, 1999
• Kreyszig, Erwin: Introductory Functional Analysis with Applications, Wiley, 1989.
• Lax, P.: Functional Analysis, Wiley-Interscience, 2002
• Lebedev, L.P. and Vorovich, I.I.: Functional Analysis in Mechanics, Springer-Verlag, 2002
• Michel, Anthony N. and Charles J. Herget: Applied Algebra and Functional Analysis, Dover, 1993.
• Reed M., Simon B. - "Functional Analysis", Academic Press 1980.
• Riesz, F. and Sz.-Nagy, B.: Functional Analysis, Dover Publications, 1990
• Rudin, W.: Functional Analysis, McGraw-Hill Science, 1991
• Schechter, M.: Principles of Functional Analysis, AMS, 2nd edition, 2001
• Shilov, Georgi E.: Elementary Functional Analysis, Dover, 1996.
• Sobolev, S.L.: Applications of Functional Analysis in Mathematical Physics, AMS, 1963
• Yosida, K.: Functional Analysis, Springer-Verlag, 6th edition, 1980

# Wikibooks

Up to date as of January 23, 2010
(Redirected to Functional Analysis article)

### From Wikibooks, the open-content textbooks collection

Functional Analysis can mean different things, depending on who you ask. The core of the subject, however, is to study linear spaces with some topology which allows us to do analysis; ones like spaces of functions, spaces of operators acting on the space of functions, etc. Our interest in those spaces is twofold: those linear spaces with topology (i) often exhibit interesting properties that are worth investigating for their own sake, and (ii) have important application in other areas of mathematics (e.g., partial differential equations) as well as theoretical physics; in particular, quantum mechanics. (i) arises because linear vectors spaces that are of interest to analysts are infinite-dimensional in nature, and this requires careful investigation of geometry. (More on this in Chapter 2 and 4.) (ii) was what initially motivated the development of the field; Functional Analysis has its historical roots in linear algebra and the mathematical formulation of quantum mechanics in the early 20 century. (See w:Mathematical formulation of quantum mechanics) The book aims to cover these two interests simultaneously.

The book consists of two parts. The first part covers the basics of Banach spaces theory with the emphasis on its applications. The second part covers topological vector spaces, especially locally convex ones, generalization of Banach spaces. In both parts, we give principal results e.g., the closed graph theorem, resulting in some repetition. One reason for doing this organization is that one often only needs a Banach-version of such results. Another reason is that this approach seems more pedagogically sound; the statement of the results in their full generality may obscure its simplicity. Exercises are meant to be unintegrated part of the book. They can be skipped altogether, and the book should be fully read and understood. Some alternative proofs and additional results are relegated as exercises when their inclusion may disrupt the flow of the exposition.

Knowledge of measure theory will not be needed except for Chapter 6, where we formulate the spectrum theorem in the language of measure theory. As for topology, knowledge of metric spaces suffices for Chapter 1 and Chapter 2. The solid background in general topology is required for the ensuing chapters.

## Contents

Part 1:

Chapter 1. Preliminaries (October, 2009)
Zorn's lemma, Topology, Hamel basis, Hahn-Banach theorem
Chapter 2. Banach spaces (Sep 1, 2007)
Open mapping theorem, closed graph theorem, compact operators
Chapter 3. Hilbert spaces (June 4, 2008)
Unbounded operators, adjoint operators, orthonormal basis, Parseval theorem
Chapter 4. Geometry of Banach spaces (May 27, 2008)
Reflexive spaces, Krein-Milman theorem, Bishop's theorem, Separable Banach spaces, Schauder basis, James' theorem, uniformly convex spaces, monotonic operators, strictly singular operators

Part 2:

Chapter 5. Topological vector spaces (May 28, 2008)
Locally convex spaces, metrization theorem
Chapter 6. C*-algebras (October 30, 2008)
Gelfand transformation, Spectrum of a commutative Banach algebra, Functional calculus, GNS construction
Chapter 7. Integration theory (August 16, 2009)
von Neumann double centralizer theorem

Part 3:

Chapter 8. Special topics (June 6, 2008)