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.
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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 can be completely classified: there is a unique Hilbert space up to isomorphism for every cardinality of the base. Since finitedimensional Hilbert spaces are fully understood in linear algebra, and since morphisms of Hilbert spaces can always be divided into morphisms of spaces with Alephnull (ℵ_{0}) dimensionality, functional analysis of Hilbert spaces mostly deals with the unique Hilbert space of dimensionality Alephnull, 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.
General Banach spaces are more complicated. There is no clear definition of what would constitute a base, for example.
Examples of Banach spaces are L^{p}spaces, for any real number (see L^{p} spaces). Given and a set X (which may be countable or uncountable), L^{p}(X) consists of "all Lebesguemeasurable functions whose absolute value's pth power has finite integral".
That is, it consists of all Lebesguemeasurable functions f for which .
If X is countable, the integral may be replaced with a sum: , although for countable X, the space is usually denoted l^{p}(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.
Important results of functional analysis include:
See also: List of functional analysis topics.
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 HahnBanach 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.
Functional analysis in its present form includes the following tendencies:

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 infinitedimensional 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 Banachversion 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.
Part 1:
Part 2:
Part 3:
