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
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.
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
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.
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
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: