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Area of mathematics
One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.
In modern introductory texts on functional analysis, the subject is seen as the study of vector spaces endowed with a topology, in particular infinite-dimensional spaces. In contrast, linear algebra deals mostly with finite-dimensional spaces, and 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.
Hilbert spaces can be completely classified: there is a unique Hilbert space up toisomorphism for every cardinality of the orthonormal basis. Finite-dimensional Hilbert spaces are fully understood in linear algebra, and infinite-dimensional separable Hilbert spaces are isomorphic to . Separability being important for applications, functional analysis of Hilbert spaces consequently mostly deals with this space. 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 than Hilbert spaces, and cannot be classified in such a simple manner as those. In particular, many Banach spaces lack a notion analogous to an orthonormal basis.
Examples of Banach spaces are -spaces for any real number Given also a measure on set then sometimes also denoted or has as its vectors equivalence classes of measurable functions whose absolute value's -th power has finite integral; that is, functions for which one has
If is the counting measure, then the integral may be replaced by a sum. That is, we require
Then it is not necessary to deal with equivalence classes, and the space is denoted written more simply in the case when is the set of non-negative integers.
In Banach spaces, a large part of the study involves the dual space: the space of all continuous linear maps from the space into its underlying field, so-called functionals. A Banach space can be canonically identified with a subspace of its bidual, which is the dual of its dual space. The corresponding map is an isometry but in general not onto. A general Banach space and its bidual need not even be isometrically isomorphic in any way, contrary to the finite-dimensional situation. 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.
Linear functional analysis
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Theorem (Uniform Boundedness Principle). Let be a Banach space and be a normed vector space. Suppose that is a collection of continuous linear operators from to . If for all in one has
There are many theorems known as the spectral theorem, but one in particular has many applications in functional analysis.
Spectral theorem. Let be a bounded self-adjoint operator on a Hilbert space . Then there is a measure space and a real-valued essentially bounded measurable function on and a unitary operator such that
Open mapping theorem. If and are Banach spaces and is a surjective continuous linear operator, then is an open map (that is, if is an open set in , then is open in ).
The proof uses the Baire category theorem, and completeness of both and is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if and are taken to be Fréchet spaces.