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.
is a non-empty subset of a topological vector space
be a complete semi-metrizable locally convex topological vector space and
be a closed convex multifunction with non-empty domain.
is a barrelled space for some/every
belongs to the relative interior of
{\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}
Closed graph theorem — Let
be Fréchet spaces and
For the non-trivial direction, assume that the graph of
is closed and convex and that its image is
Uniform boundedness principle — Let
be Fréchet spaces and
be a bijective linear map.
is an isomorphism of Fréchet spaces.
Apply the closed graph theorem to
Open mapping theorem — Let
be a continuous surjective linear map.
is a closed and convex relation whose image is
be a non-empty open subset of
The following notation and notions are used for these corollaries, where
is a non-empty subset of a topological vector space
be a barreled first countable space and let
is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that
is lower ideally convex.
belongs to the relative interior of
{\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}
be a multimap with non-empty domain.
is a barreled space, the graph of
) denote the closed unit ball in