Ursescu theorem

In mathematics, particularly in functional analysis and convex analysis, the Ursescu theorem is a theorem that generalizes the closed graph theorem, the open mapping theorem, and the uniform boundedness principle.

Ursescu Theorem

The following notation and notions are used, where is a multifunction and S is a non-empty subset of a topological vector space X:

  • the affine span of S is denoted by and the linear span is denoted by .
  • denotes the algebraic interior of S in X.
  • denotes the relative algebraic interior of S (i.e. the algebraic interior of S in ).
  • if is barreled for some/every while otherwise.
    • If S is convex then it can be shown that for any x in X, if and only if the cone generated by is a barreled linear subspace of X or equivalently, if and only if is a barreled linear subspace of X
  • The domain of is .
  • The image of is . For any subset , .
  • The graph of is .
  • is closed (respectively, convex) if the graph of is closed (resp. convex) in .
    • Note that is convex if and only if for all and all , .
  • The inverse of is the multifunction defined by . For any subset , .
    • Note that if is a function, then its inverse is the multifunction obtained from canonically identifying f with the multifunction f : X Y defined by .
  • is the topological interior of S with respect to T, where .
  • is the interior of S with respect to .

Statement

Theorem[1] (Ursescu)  Let X be a complete semi-metrizable locally convex topological vector space and be a closed convex multifunction with non-empty domain. Assume that is barreled for some/every . Assume that and let (so that ). Then for every neighborhood U of in X, belongs to the relative interior of in (i.e. ). In particular, if then .

Corollaries

Closed graph theorem

(Closed graph theorem) Let X and Y be Fréchet spaces and T : X → Y be a linear map. Then T is continuous if and only if the graph of T is closed in .

Proof: For the non-trivial direction, assume that the graph of T is closed and let . It is easy to see that is closed and convex and that its image is X. Given x in X, (T x, x) belongs to so that for every open neighborhood V of T x in Y, is a neighborhood of x in X. Thus T is continuous at x. Q.E.D.

Uniform boundedness principle

(Uniform boundedness principle) Let X and Y be Fréchet spaces and be a bijective linear map. Then T is continuous if and only if is continuous. Furthermore, if T is continuous then T is an isomorphism of Fréchet spaces.

Proof: Apply the closed graph theorem to T and . Q.E.D.

Open mapping theorem

(Open mapping theorem) Let X and Y be Fréchet spaces and be a continuous surjective linear map. Then T is an open map.

Proof: Clearly, T is a closed and convex relation whose image is Y. Let U be a non-empty open subset of X, let y be in T(U), and let x in U be such that y = T x. From the Ursescu theorem it follows that T(U) is a neighborhood of y. Q.E.D.

Additional corollaries

The following notation and notions are used for these corollaries, where is a multifunction, S is a non-empty subset of a topological vector space X:

  • a convex series with elements of S is a series of the form where all and is a series of non-negative numbers. If converges then the series is called convergent while if is bounded then the series is called bounded and b-convex.
  • S is ideally convex if any convergent b-convex series of elements of S has its sum in S.
  • S is lower ideally convex if there exists a Fréchet space Y such that S is equal to the projection onto X of some ideally convex subset B of . Every ideally convex set is lower ideally convex.

Corollary Let X be a barreled first countable space and let C be a subset of X. Then:

  1. If C is lower ideally convex then .
  2. If C is ideally convex then .

Simons' theorem

Theorem (Simons)[2] Let X and Y be first countable with X locally convex. Suppose that is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that X is a Fréchet space and that is lower ideally convex. Assume that is barreled for some/every . Assume that and let . Then for every neighborhood U of in X, belongs to the relative interior of in (i.e. ). In particular, if then .

Robinson–Ursescu theorem

The implication (1) (2) in the following theorem is known as the Robinson–Ursescu theorem.[3]

Theorem: Let and be normed spaces and be a multimap with non-empty domain. Suppose that Y is a barreled space, the graph of verifies condition condition (Hwx), and that . Let (resp. ) denote the closed unit ball in X (resp. Y) (so ). Then the following are equivalent:

  1. belongs to the algebraic interior of .
  2. .
  3. There exists such that for all , .
  4. There exist and such that for all and all , .
  5. There exists such that for all and all , .

See also

Notes

    1. Zalinescu 2002, p. 23.
    2. Zalinescu 2002, p. 22-23.
    3. Zalinescu 2002, p. 24.

    References

    • Zalinescu, C (2002). Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. ISBN 981-238-067-1. OCLC 285163112.CS1 maint: ref=harv (link)
    • Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.
    This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.