Functional analysis

• 1. Introduction

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  Functional analysis asks what happens to mathematics when a space stops having a finite number of dimensions. It is a branch of mathematical analysis built around vector spaces that carry some kind of limit-related structure, such as an inner product, a norm, or a topology. On top of those spaces sit linear functions that respect the structure underneath them. The word at its center, functional, simply means a function whose argument is itself a function. That single idea reshaped how mathematicians treat transformations like the Fourier transform, which can be seen as an operator carrying one function space into another. Why did this viewpoint prove so useful for differential and integral equations? Who first dared to treat infinite-dimensional spaces as objects in their own right? And what does it mean to say that one Hilbert space can be classified completely while general Banach spaces resist any such tidy ordering? The answers run through Italian and Polish mathematicians, through four foundational theorems, and through an invariant subspace problem that remains open.

• 2. Naming The Functional

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  The word functional, used as a noun, traces back to the calculus of variations, where it names a function whose argument is a function. The term was first used in Hadamard's 1910 book on that subject. The underlying concept, though, came earlier. In 1887 the Italian mathematician and physicist Vito Volterra introduced the general notion of a functional. The theory of nonlinear functionals was then carried forward by students of Hadamard, in particular Fréchet and Lévy. Hadamard also founded the modern school of linear functional analysis. That school was further developed by Riesz and by the group of Polish mathematicians gathered around Stefan Banach.

• 3. Into Infinite Dimensions

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  Modern introductory texts treat functional analysis as the study of vector spaces endowed with a topology, with a particular focus on infinite-dimensional spaces. Linear algebra, by contrast, deals mostly with finite-dimensional spaces and does not use topology at all. That difference is the whole point of the field. An important part of the subject extends the theories of measure, integration, and probability into infinite-dimensional spaces. This extension is also known as infinite dimensional analysis. Carrying these tools beyond the finite case is what forces the limit-related structures into the foreground.

• 4. Banach And Hilbert Spaces

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  Complete normed vector spaces over the real or complex numbers form the basic and historically first class of spaces studied in the field. Such spaces are called Banach spaces. A Hilbert space is an important example, one where the norm arises from an inner product. These spaces matter across many areas, including the mathematical formulation of quantum mechanics, machine learning, partial differential equations, and Fourier analysis. More broadly, functional analysis also takes in Fréchet spaces and other topological vector spaces that carry no norm at all. Continuous linear operators defined on Banach and Hilbert spaces are a central object of study. Those operators lead naturally to the definition of C*-algebras and other operator algebras.

• 5. The Classifiable Space

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  Hilbert spaces can be completely classified, which sets them apart from most objects in the field. Up to isomorphism, there is a unique Hilbert space for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are already fully understood within linear algebra. Separability matters for applications, so functional analysis of Hilbert spaces mostly deals with the separable infinite-dimensional case. One open problem in the field 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.

• 6. The Unruly Space

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  General Banach spaces are more complicated than Hilbert spaces and cannot be classified in any comparably simple way. Many of them lack any notion analogous to an orthonormal basis. Much of the study of these spaces involves the dual space, the space of all continuous linear maps from the space into its underlying field, which are the functionals. A Banach space can be canonically identified with a subspace of its bidual, the dual of its dual space. The corresponding map is an isometry, but in general it is not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. The notion of derivative can also be extended to arbitrary functions between Banach spaces, as in the Fréchet derivative.

• 7. The Four Pillars

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  Four major theorems are sometimes called the four pillars of functional analysis. They are the Hahn-Banach theorem, the open mapping theorem, the closed graph theorem, and the uniform boundedness principle, also known as the Banach-Steinhaus theorem. The uniform boundedness principle was first published in 1927 by Stefan Banach and Hugo Steinhaus, and it was also proven independently by Hans Hahn. In its basic form, it says that for a family of continuous linear operators whose domain is a Banach space, pointwise boundedness is equivalent to uniform boundedness in operator norm. The Hahn-Banach theorem extends bounded linear functionals defined on a subspace to the whole space, and it shows there are enough continuous linear functionals on every normed vector space to make the dual space interesting. The open mapping theorem, also known as the Banach-Schauder theorem after Stefan Banach and Juliusz Schauder, states that a surjective continuous linear operator between Banach spaces is an open map. Its proof uses the Baire category theorem, and completeness is essential; the statement remains true for Fréchet spaces but fails for spaces that are merely normed.

• 8. Foundations And Directions

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  Most spaces in functional analysis have infinite dimension, which raises questions reaching down to the foundations of mathematics. Showing that such a space has a vector space basis may require Zorn's lemma, though the Schauder basis is usually the more relevant concept. Proving the Hahn-Banach theorem typically uses the axiom of choice, although the strictly weaker Boolean prime ideal theorem suffices, and the Baire category theorem also requires a form of the axiom of choice. The field branches into several tendencies. Abstract analysis works through topological groups, topological rings, and topological vector spaces. The geometry of Banach spaces includes a combinatorial approach connected with Jean Bourgain and questions about when forms of the law of large numbers hold. Noncommutative geometry was developed by Alain Connes, building partly on George Mackey's approach to ergodic theory. The connection with quantum mechanics was interpreted broadly by Israel Gelfand to take in most types of representation theory.