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.  
  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

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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.