I apologize in advance for the rather vague question.
While reading the book White noise distribution theory by H.H. Kuo, in particular the section 13.3 I came across the following statement (I'll paraphrase it)
Since the Hida space $(S)^*$ is not Banach we cannot simply use the standard definition of the Bochner integral [...]. However note that $(S)^*=\bigcup_{p\geq 0} (S_{-p})^*$ and each $(S_{-p})^*$ is a separable Hilbert space. Then it's reasonable to say that a function $\Phi:M\to (S)^*$ is Bochner integrable if:
- $\Phi$ is weakly measurable.
- $\exists p\geq 0$ such that $\Phi(u)\in (S_{-p})^*$ for a.a. $u\in M$ and $\|\Phi(\cdot)\|_{-p}\in L^1(M)$.
Since the space of tempered distributions $S'(\mathbb R)$ can also be constructed in a similar fashion $S'(\mathbb R)=\bigcup_{p\geq 0} S_{-p}(\mathbb R)$ where $S_{-p}(\mathbb R)$ is the completition of $L^2(\mathbb R)$ with respect to the norm $|f|_{-p}:=\|A^{-p}f\|$ and $A$ is the Hamiltonian of the harmonic oscillator.
Can we extend the definition of Bochner integrals to functions taking values in the space of tempered distributions?
