Ursescu theorem

Template:Short description 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

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

• the affine span of $S$ is denoted by $\mathrm{aff}S$ and the linear span is denoted by $\mathrm{span}S.$
• $S^i:={\mathrm{aint}}_{X}S$ denotes the algebraic interior of $S$ in $X.$
• ${}^{i}S:={\mathrm{aint}}_{\mathrm{aff}(S-S)}S$ denotes the relative algebraic interior of $S$ (i.e. the algebraic interior of $S$ in $\mathrm{aff}(S-S)$).
• ${}^{ib}S:={}^{i}S$ if $\mathrm{span}(S-s_0)$ is barreled for some/every $s_0\in S$ while ${}^{ib}S:=\varnothing$ otherwise.
  • If $S$ is convex then it can be shown that for any $x\in X,$ $x\in {}^{ib}S$ if and only if the cone generated by $S-x$ is a barreled linear subspace of $X$ or equivalently, if and only if ${\cup }_{n\in \mathbb{N}}n(S-x)$ is a barreled linear subspace of $X$
• The domain of $ℛ$ is $\mathrm{Dom}ℛ:=\{x\in X:ℛ(x)\ne \varnothing \}.$
• The image of $ℛ$ is $\mathrm{Im}ℛ:={\cup }_{x\in X}ℛ(x).$ For any subset $A\subseteq X,$ $ℛ(A):={\cup }_{x\in A}ℛ(x).$
• The graph of $ℛ$ is $\mathrm{gr}ℛ:=\{(x,y)\in X\times Y:y\in ℛ(x)\}.$
• $ℛ$ is closed (respectively, convex) if the graph of $ℛ$ is closed (resp. convex) in $X\times Y.$
  • Note that $ℛ$ is convex if and only if for all $x_0,x_1\in X$ and all $r\in [0,1],$ $rℛ\left(x_0\right)+(1-r)ℛ\left(x_1\right)\subseteq ℛ(r{x}_0+(1-r)x_1).$
• The inverse of $ℛ$ is the set-valued function ${ℛ}^{-1}:Y\rightrightarrows X$ defined by ${ℛ}^{-1}(y):=\{x\in X:y\in ℛ(x)\}.$ For any subset $B\subseteq Y,$ ${ℛ}^{-1}(B):={\cup }_{y\in B}{ℛ}^{-1}(y).$
  • If $f:X\to Y$ is a function, then its inverse is the set-valued function $f^{-1}:Y\rightrightarrows X$ obtained from canonically identifying $f$ with the set-valued function $f:X\rightrightarrows Y$ defined by $x\mapsto \{f(x)\}.$
• ${\mathrm{int}}_{T}S$ is the topological interior of $S$ with respect to $T,$ where $S\subseteq T.$
• $\mathrm{rint}S:={\mathrm{int}}_{\mathrm{aff}S}S$ is the interior of $S$ with respect to $\mathrm{aff}S.$

Corollaries

Additional corollaries

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

• a convex series with elements of $S$ is a series of the form ${\sum }_{i=1}^{\mathrm{\infty }}{r}_i{s}_i$ where all $s_i\in S$ and ${\sum }_{i=1}^{\mathrm{\infty }}{r}_i=1$ is a series of non-negative numbers. If ${\sum }_{i=1}^{\mathrm{\infty }}{r}_i{s}_i$ converges then the series is called convergent while if ${\left(s_i\right)}_{i=1}^{\mathrm{\infty }}$ 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 space $Y$ such that $S$ is equal to the projection onto $X$ of some ideally convex subset B of $X\times Y.$ Every ideally convex set is lower ideally convex.

Related theorems

Robinson–Ursescu theorem

The implication (1) $⟹$ (2) in the following theorem is known as the Robinson–Ursescu theorem.Template:Sfn

See also

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References

• Template:Zălinescu Convex Analysis in General Vector Spaces 2002
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