F. Riesz's theorem (named after Frigyes Riesz) is an important theorem in

functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...

that states that a Hausdorff

topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...

(TVS) is finite-dimensional if and only if it is

locally compact In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which ev ...

. The theorem and its consequences are used ubiquitously in functional analysis, often used without being explicitly mentioned.

Statement

Recall that a

(TVS)

X

is Hausdorff if and only if the singleton set

\

consisting entirely of the origin is a closed subset of

X.

A map between two TVSs is called a TVS-isomorphism or an isomorphism in the category of TVSs if it is a

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...

Consequences

Throughout,

F, X, Y

are TVSs (not necessarily Hausdorff) with

F

a finite-dimensional vector space. * Every finite-dimensional vector subspace of a Hausdorff TVS is a closed subspace. * All finite-dimensional Hausdorff TVSs are

Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...

s and all norms on such a space are equivalent. * Closed + finite-dimensional is closed: If

M

is a closed vector subspace of a TVS

Y

and if

F

is a finite-dimensional vector subspace of

Y

(

Y, M,

and

F

are not necessarily Hausdorff) then

M + F

is a closed vector subspace of

Y.

* Every vector space isomorphism (i.e. a

bijection In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...

) between two finite-dimensional Hausdorff TVSs is a

TVS isomorphism In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is al ...

. * Uniqueness of topology: If

X

is a finite-dimensional vector space and if

\tau_1

and

\tau_2

are two Hausdorff TVS topologies on

X

then

\tau_1 = \tau_2.

* Finite-dimensional domain: A

linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...

L : F \to Y

between Hausdorff TVSs is necessarily continuous. ** In particular, every

linear functional In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the ...

of a finite-dimensional Hausdorff TVS is continuous. * Finite-dimensional range: Any continuous

surjective In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...

L : X \to Y

with a Hausdorff finite-dimensional range is an

open map In mathematics, more specifically in topology, an open map is a function between two topological spaces that maps open sets to open sets. That is, a function f : X \to Y is open if for any open set U in X, the image f(U) is open in Y. Likewise, ...

and thus a

topological homomorphism In functional analysis, a topological homomorphism or simply homomorphism (if no confusion will arise) is the analog of homomorphisms for the category of topological vector spaces (TVSs). This concept is of considerable importance in functional ana ...

. In particular, the range of

L

is TVS-isomorphic to

X / L^(0).

* A TVS

X

(not necessarily Hausdorff) is locally compact if and only if

X / \overline

is finite dimensional. * The

convex hull In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space ...

of a compact subset of a finite-dimensional Hausdorff TVS is compact. ** This implies, in particular, that the convex hull of a compact set is equal to the convex hull of that set. * A Hausdorff

locally bounded In mathematics, a function is locally bounded if it is bounded around every point. A family of functions is locally bounded if for any point in their domain all the functions are bounded around that point and by the same number. Locally bounded ...

TVS with the Heine-Borel property is necessarily finite-dimensional.

References

Bibliography

* * * * {{Functional analysis Functional analysis Lemmas

Statement

Consequences

See also

References

Bibliography