This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.

A lot of the time in infinite-dimensional stochastic analysis one deals with processes that take values in some Banach space of functions and that is continuously embedded in the Schwartz distributions. This motivates the following observations and questions.

Let $V$ be a nuclear Fréchet space. (EDIT: It would probably make more sense to assume that $V$ is the dual of a nuclear Fréchet space, like the Schwartz distributions.) With the probability space $(\Omega,\mathscr{F},\mathbb{P})$, let $X\in V\hat{\otimes}L^1(\mathbb{P})$ and $\mathscr{G}\subset\mathscr{F}$ be a sub-$\sigma$-algebra. Now the conditional expectation $\mathbb{E}[\cdot\vert\mathscr{G}]$ extends to $V\hat{\otimes}L^1(\mathbb{P})$ by tensor product properties (I think, I'm still a bit shaky around non-Hilbert tensor products). With conditional expectations in hand we can define martingales etc. We may also say that the quadratic covariation of two processes $X,Y$ is (if it exists) a $V\hat\otimes V$-valued process such that $$ X\otimes Y - \langle X,Y\rangle $$ is a martingale. However, I don't think we will have a scalar quadratic variation since that usually requires a Hilbert space. So suppose $H\hookrightarrow V$ is a densely embedded Hilbert space and $X,Y$ take values in it. What, if any, is the relation between the quadratic covariation described above and the usual Hilbert space one? How about Brownian motion, stochastic integrals, the Itô isometry...?

In the literature I know of this paper and the references therein which deal with topics in this area and also this chapter. But it all strikes me as somewhat disparate. Are there no textbook references for this material?