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Ursescu theorem

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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

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The following notation and notions are used, where is a set-valued function and is a non-empty subset of a topological vector space :

  • the affine span of is denoted by and the linear span is denoted by
  • denotes the algebraic interior of in
  • denotes the relative algebraic interior of (i.e. the algebraic interior of in ).
  • if is barreled for some/every while otherwise.
    • If is convex then it can be shown that for any if and only if the cone generated by is a barreled linear subspace of or equivalently, if and only if is a barreled linear subspace of
  • 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 set-valued function defined by For any subset
    • If is a function, then its inverse is the set-valued function obtained from canonically identifying with the set-valued function defined by
  • is the topological interior of with respect to where
  • is the interior of with respect to

Statement

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

Corollaries

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Closed graph theorem

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Closed graph theorem — Let and be Fréchet spaces and be a linear map. Then is continuous if and only if the graph of is closed in

Proof

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

Uniform boundedness principle

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

Proof

Apply the closed graph theorem to and Q.E.D.

Open mapping theorem

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Open mapping theorem — Let and be Fréchet spaces and be a continuous surjective linear map. Then T is an open map.

Proof

Clearly, is a closed and convex relation whose image is Let be a non-empty open subset of let be in and let in be such that From the Ursescu theorem it follows that is a neighborhood of Q.E.D.

Additional corollaries

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The following notation and notions are used for these corollaries, where is a set-valued function, is a non-empty subset of a topological vector space :

  • a convex series with elements of 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.
  • is ideally convex if any convergent b-convex series of elements of has its sum in
  • is lower ideally convex if there exists a Fréchet space such that is equal to the projection onto of some ideally convex subset B of Every ideally convex set is lower ideally convex.

Corollary — Let be a barreled first countable space and let be a subset of Then:

  1. If is lower ideally convex then
  2. If is ideally convex then
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Simons' theorem

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Simons' theorem[2] — Let and be first countable with locally convex. Suppose that is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that 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 of in belongs to the relative interior of in (i.e. ). In particular, if then

Robinson–Ursescu theorem

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

Robinson–Ursescu theorem[3] — Let and be normed spaces and be a multimap with non-empty domain. Suppose that is a barreled space, the graph of verifies condition condition (Hwx), and that Let (resp. ) denote the closed unit ball in (resp. ) (so ). Then the following are equivalent:

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

See also

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Notes

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  1. ^ Zălinescu 2002, p. 23.
  2. ^ Zălinescu 2002, p. 22-23.
  3. ^ a b Zălinescu 2002, p. 24.

References

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  • Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
  • 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.