Ursescu theorem

Short description: Generalization of closed graph, open mapping, and uniform boundedness 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 ${ℛ}:X\rightrightarrows Y$ 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 ${\cup }_{n\in \mathbb{\mathbb{N}}}n(S-x)$ 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 $r\in [0,1],$ $r{ℛ}\left(x_0\right)+(1-r){ℛ}\left(x_1\right)\subseteq {ℛ}(r{x}_0+(1-r)x_1).$
• 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 $\mathrm{aff}S.$

Statement

Theorem (Ursescu) — Let $X$ be a complete semi-metrizable locally convex topological vector space and ${ℛ}:X\rightrightarrows Y$ be a closed convex multifunction with non-empty domain. Assume that $\mathrm{span}(\mathrm{Im}{ℛ}-y)$ is a barrelled space for some/every $y\in \mathrm{Im}{ℛ}.$ Assume that $y_0\in {}^i(\mathrm{Im}{ℛ})$ and let $x_0\in {{ℛ}}^{-1}\left(y_0\right)$ (so that $y_0\in {ℛ}\left(x_0\right)$). Then for every neighborhood $U$ of $x_0$ in $X,$ $y_0$ belongs to the relative interior of ${ℛ}(U)$ in $\mathrm{aff}(\mathrm{Im}{ℛ})$ (that is, $y_0\in {\mathrm{int}}_{\mathrm{aff}(\mathrm{Im}{ℛ})}{ℛ}(U)$). In particular, if ${}^{ib}(\mathrm{Im}{ℛ})\ne \varnothing$ then ${}^{ib}(\mathrm{Im}{ℛ})={}^i(\mathrm{Im}{ℛ})=\mathrm{rint}(\mathrm{Im}{ℛ}).$

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 ${ℛ}:=T^{-1}:Y\rightrightarrows X.$ It is easy to see that $\mathrm{gr}{ℛ}$ 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 $Y,$ ${ℛ}(V)=T^{-1}(V)$ 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 ${ℛ}:X\rightrightarrows Y$ 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.

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 $C^i=\mathrm{int}C.$
2. If $C$ is ideally convex then $C^i=\mathrm{int}C=\mathrm{int}(\mathrm{cl}C)={(\mathrm{cl}C)}^i.$

Related theorems

Simons' theorem

Simons' theorem — Let $X$ and $Y$ be first countable with $X$ locally convex. Suppose that ${ℛ}:X\rightrightarrows Y$ is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that $X$ is a Fréchet space and that ${ℛ}$ is lower ideally convex. Assume that $\mathrm{span}(\mathrm{Im}{ℛ}-y)$ is barreled for some/every $y\in \mathrm{Im}{ℛ}.$ Assume that $y_0\in {}^i(\mathrm{Im}{ℛ})$ and let $x_0\in {{ℛ}}^{-1}\left(y_0\right).$ Then for every neighborhood $U$ of $x_0$ in $X,$ $y_0$ belongs to the relative interior of ${ℛ}(U)$ in $\mathrm{aff}(\mathrm{Im}{ℛ})$ (i.e. $y_0\in {\mathrm{int}}_{\mathrm{aff}(\mathrm{Im}{ℛ})}{ℛ}(U)$). In particular, if ${}^{ib}(\mathrm{Im}{ℛ})\ne \varnothing$ then ${}^{ib}(\mathrm{Im}{ℛ})={}^i(\mathrm{Im}{ℛ})=\mathrm{rint}(\mathrm{Im}{ℛ}).$

Robinson–Ursescu theorem

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

Robinson–Ursescu theorem — Let $(X,\Vert \cdot \Vert )$ and $(Y,\Vert \cdot \Vert )$ be normed spaces and ${ℛ}:X\rightrightarrows Y$ be a multimap with non-empty domain. Suppose that $Y$ is a barreled space, the graph of ${ℛ}$ verifies condition condition (Hwx), and that $(x_0,y_0)\in \mathrm{gr}{ℛ}.$ Let $C_X$ (resp. $C_Y$) denote the closed unit ball in $X$ (resp. $Y$) (so $C_X=\{x\in X:\Vert x\Vert \le 1\}$). Then the following are equivalent:

1. $y_0$ belongs to the algebraic interior of $\mathrm{Im}{ℛ}.$
2. $y_0\in \mathrm{int}{ℛ}(x_0+C_X).$
3. There exists $B>0$ such that for all $0\le r\le 1,$ $y_0+Br{C}_Y\subseteq {ℛ}(x_0+r{C}_X).$
4. There exist $A>0$ and $B>0$ such that for all $x\in x_0+A{C}_X$ and all $y\in y_0+A{C}_Y,$ $d(x,{{ℛ}}^{-1}(y))\le B\cdot d(y,{ℛ}(x)).$
5. There exists $B>0$ such that for all $x\in X$ and all $y\in y_0+B{C}_Y,$ $d(x,{{ℛ}}^{-1}(y))\le \frac{1+\Vert x-x_0\Vert }{B-\Vert y-y_0\Vert }\cdot d(y,{ℛ}(x)).$

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 – Theorem stating that pointwise boundedness implies uniform boundedness
• Webbed space – Space where open mapping and closed graph theorems hold

References

• Template:Zălinescu Convex Analysis in General Vector Spaces 2002
• Baggs, Ivan (1974). "Functions with a closed graph" (in en). Proceedings of the American Mathematical Society 43 (2): 439–442. doi:10.1090/S0002-9939-1974-0334132-8. ISSN 0002-9939. https://www.ams.org/.