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Suppose that $(H, \langle\cdot,\cdot\rangle)$ is a (separable) Hilbert space, $E$ is Banach with $\mathcal{C}\equiv C_b(E;E)$ the (Banach) space of continuous $\&$ bounded $E$-valued functions, and let $S : E \rightarrow H$ and $T : \mathcal{D}\times H \rightarrow H$ be (nonlinear) continuous operators, for some $\mathcal{D}\subset\mathcal{C}$ dense.

Suppose further that for each $g\in\mathcal{D}$, we have the 'duality relation'

$$\tag{1}\langle S(g(e)), h\rangle = \langle S(e), T(g,h)\rangle \qquad \text{for each } \ h\in H.$$

Can the operator $T$ be uniquely extended to a continuous operator $\overline{T} : \mathcal{C}\times H \rightarrow H$ which still satisfies $(1)$, but for all $g\in\mathcal{C}$?

(Or are the above assumptions too generic to answer this question? Any hints or references are welcome.)

fsp-b
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    If you can show $T$ is uniformly continuous then you get a unique uniformly continuous extension for free and (1) holds by continuity. – blamethelag Dec 12 '21 at 21:52
  • Thanks for your comment, @blamethelag! Do you happen to have a reference for this? If you'd like to extend your comment (slightly more detailed maybe, assuming uniform continuity of $T$ as you wish) I'd accept it as an answer. – fsp-b Dec 12 '21 at 22:16
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    The result I am mentioning is extremly usefull but I have never found any reference, except for my handwritten lecture notes. It has been discussed a lot her, for instance https://math.stackexchange.com/questions/245237/extension-of-a-uniformly-continuous-function-between-metric-spaces?utm_medium=organic&utm_source=google_rich_qa&utm_campaign=google_rich_qa – blamethelag Dec 12 '21 at 22:48
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    @blamethelag Thanks a lot! Seems like Theorem 5.1 in "Topological Vector Spaces, Distributions and Kernels" by Francois Treves is a published reference for the result you mention. (The relevant section of that reference is available on google books.) – fsp-b Dec 12 '21 at 23:06

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