Ursescu theorem

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

Proof: For the non-trivial direction, assume that the graph of 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 X. Given x in X, (T x, x) belongs to ${\displaystyle Y\times X}$ so that for every open neighborhood V of T x in Y, ${\displaystyle {\mathcal {R}}(V)=T^{-1}(V)}$ is a neighborhood of x in X. Thus T is continuous at x. Q.E.D.

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

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

(Open mapping theorem) Let X and Y be Fréchet spaces and ${\displaystyle T:X\to Y}$ 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 = T x. From the Ursescu theorem it follows that T(U) is a neighborhood of y. Q.E.D.

The following notation and notions are used for these corollaries, where ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ is a multifunction, 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 ${\displaystyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ where all ${\displaystyle s_{i}\in S}$ and ${\displaystyle \sum _{i=1}^{\infty }r_{i}=1}$ is a series of non-negative numbers. If ${\displaystyle \sum _{i=1}^{\infty }r_{i}s_{i}}$ converges then the series is called convergent while if ${\displaystyle \left(s_{i}\right)_{i=1}^{\infty }}$ 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 ${\displaystyle X\times Y}$. 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 ${\displaystyle C^{i}=\operatorname {int} C}$.
2. If C is ideally convex then ${\displaystyle C^{i}=\operatorname {int} C=\operatorname {int} \left(\operatorname {cl} C\right)=\left(\operatorname {cl} C\right)^{i}}$.

Theorem (Simons)[2] Let X and Y be first countable with X locally convex. Suppose that ${\displaystyle {\mathcal {R}}:X\rightrightarrows Y}$ is a multimap with non-empty domain that satisfies condition (Hwx) or else assume that X is a Fréchet space and that ${\displaystyle {\mathcal {R}}}$ is lower ideally convex. Assume that ${\displaystyle \operatorname {span} \left(\operatorname {Im} {\mathcal {R}}-y\right)}$ is barreled for some/every ${\displaystyle y\in \operatorname {Im} {\mathcal {R}}}$. Assume that ${\displaystyle y_{0}\in {}^{i}\left(\operatorname {Im} {\mathcal {R}}\right)}$ and let ${\displaystyle x_{0}\in {\mathcal {R}}^{-1}\left(y_{0}\right)}$. Then for every neighborhood U of ${\displaystyle x_{0}}$ in X, ${\displaystyle y_{0}}$ belongs to the relative interior of ${\displaystyle {\mathcal {R}}\left(U\right)}$ in ${\displaystyle \operatorname {aff} \left(\operatorname {Im} {\mathcal {R}}\right)}$ (i.e. ${\displaystyle y_{0}\in \operatorname {int} _{\operatorname {aff} \left(\operatorname {Im} {\mathcal {R}}\right)}{\mathcal {R}}\left(U\right)}$). In particular, if ${\displaystyle {}^{ib}\left(\operatorname {Im} {\mathcal {R}}\right)\neq \emptyset }$ then ${\displaystyle {}^{ib}\left(\operatorname {Im} {\mathcal {R}}\right)={}^{i}\left(\operatorname {Im} {\mathcal {R}}\right)=\operatorname {rint} \left(\operatorname {Im} {\mathcal {R}}\right)}$.

The implication (1) ${\displaystyle \implies }$ (2) in the following theorem is known as the Robinson–Ursescu theorem.[3]

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

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