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 .
Theorem[1](Ursescu) — Let X be a completesemi-metrizablelocally convextopological vector space and be a closedconvex 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.
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 spaceX:
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 spaceY 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:
If C is lower ideally convex then .
If C is ideally convex then .
Related theorems
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:
Zalinescu, C (2002). Convex analysis in general vector spaces. River Edge, N.J. London: World Scientific. ISBN981-238-067-1. OCLC285163112. {{cite book}}: Invalid |ref=harv (help)