Ursescu theorem

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 ${\mathcal {R}}:X\rightrightarrows Y$ 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 $\operatorname {aff} S$ and the linear span is denoted by $\operatorname {span} S.$
• $S^{i}:=\operatorname {aint} _{X}S$ denotes the algebraic interior of $S$ in $X.$
• ${}^{i}S:=\operatorname {aint} _{\operatorname {aff} (S-S)}S$ denotes the relative algebraic interior of $S$ (i.e. the algebraic interior of $S$ in $\operatorname {aff} (S-S)$).
• ${}^{ib}S:={}^{i}S$ if $\operatorname {span} \left(S-s_{0}\right)$ 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 ${\mathcal {R}}$ is $\operatorname {Dom} {\mathcal {R}}:=\{x\in X:{\mathcal {R}}(x)\neq \varnothing \}.$
• The image of ${\mathcal {R}}$ is $\operatorname {Im} {\mathcal {R}}:=\cup _{x\in X}{\mathcal {R}}(x).$ For any subset $A\subseteq X,$ ${\mathcal {R}}(A):=\cup _{x\in A}{\mathcal {R}}(x).$
• The graph of ${\mathcal {R}}$ is $\operatorname {gr} {\mathcal {R}}:=\{(x,y)\in X\times Y:y\in {\mathcal {R}}(x)\}.$
• ${\mathcal {R}}$ is closed (respectively, convex) if the graph of ${\mathcal {R}}$ is closed (resp. convex) in $X\times Y.$
  • Note that ${\mathcal {R}}$ is convex if and only if for all $x_{0},x_{1}\in X$ and all $r\in [0,1],$ $r{\mathcal {R}}\left(x_{0}\right)+(1-r){\mathcal {R}}\left(x_{1}\right)\subseteq {\mathcal {R}}\left(rx_{0}+(1-r)x_{1}\right).$
• The inverse of ${\mathcal {R}}$ is the set-valued function ${\mathcal {R}}^{-1}:Y\rightrightarrows X$ defined by ${\mathcal {R}}^{-1}(y):=\{x\in X:y\in {\mathcal {R}}(x)\}.$ For any subset $B\subseteq Y,$ ${\mathcal {R}}^{-1}(B):=\cup _{y\in B}{\mathcal {R}}^{-1}(y).$
  • 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 $x\mapsto \{f(x)\}.$
• $\operatorname {int} _{T}S$ is the topological interior of $S$ with respect to $T,$ where $S\subseteq T.$
• $\operatorname {rint} S:=\operatorname {int} _{\operatorname {aff} S}S$ is the interior of $S$ with respect to $\operatorname {aff} S.$

Statement

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

Corollaries

Closed graph theorem

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

Proof

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

Uniform boundedness principle

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

Proof

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

Open mapping theorem

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

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 $y=Tx.$ 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 ${\mathcal {R}}:X\rightrightarrows Y$ 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 ${\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ where all $s_{i}\in S$ and ${\textstyle \sum _{i=1}^{\infty }r_{i}=1}$ is a series of non-negative numbers. If ${\textstyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ converges then the series is called convergent while if $\left(s_{i}\right)_{i=1}^{\infty }$ 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.

Corollary — Let $X$ be a barreled first countable space and let $C$ be a subset of $X.$ Then:

1. If $C$ is lower ideally convex then $C^{i}=\operatorname {int} C.$
2. If $C$ is ideally convex then $C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}.$

Related theorems

Simons' theorem

Simons' theorem — Let $X$ and $Y$ be first countable with $X$ locally convex. Suppose that ${\mathcal {R}}:X\rightrightarrows Y$ is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that $X$ is a Fréchet space and that ${\mathcal {R}}$ is lower ideally convex. Assume that $\operatorname {span} (\operatorname {Im} {\mathcal {R}}-y)$ is barreled for some/every $y\in \operatorname {Im} {\mathcal {R}}.$ Assume that $y_{0}\in {}^{i}(\operatorname {Im} {\mathcal {R}})$ and let $x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right).$ Then for every neighborhood $U$ of $x_{0}$ in $X,$ $y_{0}$ belongs to the relative interior of ${\mathcal {R}}(U)$ in $\operatorname {aff} (\operatorname {Im} {\mathcal {R}})$ (i.e. $y_{0}\in \operatorname {int} _{\operatorname {aff} (\operatorname {Im} {\mathcal {R}})}{\mathcal {R}}(U)$). In particular, if ${}^{ib}(\operatorname {Im} {\mathcal {R}})\neq \varnothing$ then ${}^{ib}(\operatorname {Im} {\mathcal {R}})={}^{i}(\operatorname {Im} {\mathcal {R}})=\operatorname {rint} (\operatorname {Im} {\mathcal {R}}).$

Robinson–Ursescu theorem

The implication (1) $\implies$ (2) in the following theorem is known as the Robinson–Ursescu theorem.

Robinson–Ursescu theorem — Let $(X,\|\,\cdot \,\|)$ and $(Y,\|\,\cdot \,\|)$ be normed spaces and ${\mathcal {R}}:X\rightrightarrows Y$ be a multimap with non-empty domain. Suppose that $Y$ is a barreled space, the graph of ${\mathcal {R}}$ verifies condition condition (Hwx), and that $(x_{0},y_{0})\in \operatorname {gr} {\mathcal {R}}.$ Let $C_{X}$ (resp. $C_{Y}$) denote the closed unit ball in $X$ (resp. $Y$) (so $C_{X}=\{x\in X:\|x\|\leq 1\}$). Then the following are equivalent:

1. $y_{0}$ belongs to the algebraic interior of $\operatorname {Im} {\mathcal {R}}.$
2. $y_{0}\in \operatorname {int} {\mathcal {R}}\left(x_{0}+C_{X}\right).$
3. There exists $B>0$ such that for all $0\leq r\leq 1,$ $y_{0}+BrC_{Y}\subseteq {\mathcal {R}}\left(x_{0}+rC_{X}\right).$
4. There exist $A>0$ and $B>0$ such that for all $x\in x_{0}+AC_{X}$ and all $y\in y_{0}+AC_{Y},$ $d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq B\cdot d(y,{\mathcal {R}}(x)).$
5. There exists $B>0$ such that for all $x\in X$ and all $y\in y_{0}+BC_{Y},$ $d\left(x,{\mathcal {R}}^{-1}(y)\right)\leq {\frac {1+\left\|x-x_{0}\right\|}{B-\left\|y-y_{0}\right\|}}\cdot d(y,{\mathcal {R}}(x)).$

See also

• Closed graph theorem – Theorem relating continuity to graphs
• Closed graph theorem (functional analysis) – Theorems connecting continuity to closure of graphs
• Open mapping theorem (functional analysis) – Condition for a linear operator to be open
• Surjection of Fréchet spaces – Characterization of surjectivity
• Uniform boundedness principle – A theorem stating that pointwise boundedness implies uniform boundedness
• Webbed space – Space where open mapping and closed graph theorems hold

References

• Zălinescu, Constantin (30 July 2002). Convex Analysis in General Vector Spaces. River Edge, N.J. London: World Scientific Publishing. ISBN 978-981-4488-15-0. MR 1921556. OCLC 285163112 – via Internet Archive.
• Baggs, Ivan (1974). "Functions with a closed graph". Proceedings of the American Mathematical Society. 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939.