Let $A,B,C$ be Banach spaces and $m\,:\,A\times B\to C$ be a bilinear map such that $\|m(a,b)\|\leq \textrm{const}\,\|a\|\|b\|$. We denote by $\mathcal{S}(\mathbb{R}^d)$ be the standard space of Schwartz test functions and we denote by $\mathcal{S}(\mathbb{R}^d,B)$ the space of Schwartz test functions valued in $B$.


Let $F$ be a linear and continuous functional $\mathcal{S}(\mathbb{R}^d)\to A$. For $g\in \mathcal{S}(\mathbb{R}^d,B)$ of the form $g=\sum\_{i=1}^N f\_i b\_i$, where $f\_i\in\mathcal{S}(\mathbb{R}^d)$, $b\_i\in B$ we define
$$
\langle F,g\rangle\_m := \sum\_{i=1}^N m(\langle F,f\_i\rangle,b\_i)\in C.
$$
Does the above bracket extend uniquely to all $g\in \mathcal{S}(\mathbb{R}^d,B)$?



Banach space valued distributions and test functions
Yes.


The tensor product $\mathcal S(\mathbb R^d)\otimes B$ can be identified with a subspace of $\mathcal S(\mathbb R^d,B)$ by denoting by $g\otimes b$ the function $x\mapsto g(x)b$ for all $g\in\mathcal S(\mathbb R^d)$ and $b\in B$ and this is the space you're first defining $g\mapsto\langle F,g\rangle\_m$ on.


Then, since $\mathcal S(\mathbb R^d,B)$ is the completion of $\mathcal S(\mathbb R^d)\otimes B$ for the projective tensor product topology, your question reduces to showing that the bilinear mapping


$$\begin{array}{ccc}\mathcal S(\mathbb R^d)\times B&\longrightarrow&C\\ (g,b)&\longmapsto&m(\langle F,g\rangle,b)\end{array}$$


is continuous. But this is clear since both $F$ and $m$ are continuous.


