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.

The following notation and notions are used, where ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ is a set-valued function and ${\displaystyle S}$ is a non-empty subset of a topological vector space ${\displaystyle X}$:

Theorem (Ursescu)—Let ${\displaystyle X}$ be a complete semi-metrizable locally convex topological vector space and ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ be a closed convex multifunction with non-empty domain. Assume that ${\displaystyle \operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)}$ is a barrelled space for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}.}$ Assume that ${\displaystyle y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})}$ and let ${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}$ (so that ${\displaystyle y_{0}\in {\mathcal {R}}\left(x_{0}\right)}$). Then for every neighborhood ${\displaystyle U}$ of ${\displaystyle x_{0}}$ in ${\displaystyle X,}$ ${\displaystyle y_{0}}$ belongs to the relative interior of ${\displaystyle {\mathcal {R}}(U)}$ in ${\displaystyle \operatorname {aff} (\operatorname {Im} {\mathcal {R}})}$ (that is, ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)}$). In particular, if ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing }$ then ${\displaystyle {}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).}$

Closed graph theorem—Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and ${\displaystyle T:X\to Y}$ be a linear map. Then ${\displaystyle T}$ is continuous if and only if the graph of ${\displaystyle T}$ is closed in ${\displaystyle X\times Y.}$

For the non-trivial direction, assume that the graph of ${\displaystyle T}$ is closed and let ${\displaystyle {\mathcal {R}}:=T^{-1}:Y\rightrightarrows X.}$ It is easy to see that ${\displaystyle \operatorname {gr} {\mathcal {R}}}$ is closed and convex and that its image is ${\displaystyle X.}$ Given ${\displaystyle x\in X,}$ ${\displaystyle (Tx,x)}$ belongs to ${\displaystyle Y\times X}$ so that for every open neighborhood ${\displaystyle V}$ of ${\displaystyle Tx}$ in ${\displaystyle Y,}$ ${\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}$ is a neighborhood of ${\displaystyle x}$ in ${\displaystyle X.}$ Thus ${\displaystyle T}$ is continuous at ${\displaystyle x.}$ Q.E.D.

Uniform boundedness principle—Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and ${\displaystyle T:X\to Y}$ be a bijective linear map. Then ${\displaystyle T}$ is continuous if and only if ${\displaystyle T^{-1}:Y\to X}$ is continuous. Furthermore, if ${\displaystyle T}$ is continuous then ${\displaystyle T}$ is an isomorphism of Fréchet spaces.

Apply the closed graph theorem to ${\displaystyle T}$ and ${\displaystyle T^{-1}.}$ Q.E.D.

Open mapping theorem—Let ${\displaystyle X}$ and ${\displaystyle Y}$ be Fréchet spaces and ${\displaystyle T:X\to Y}$ be a continuous surjective linear map. Then T is an open map.