I came across the following question to which I could not figure out a straightforward answer:
Let $H$ be a separable, (in general infinite-dimensional) real Hilbert space and $V$ a separable Banach space such that $V \subseteq H$ continuously and densely. $\mathbb{B}$ is defined as the following space of paths:
$$\mathbb{B} := \{y \in C(\mathbb{R}_+,H)|\int_{0}^{T}||y(s)||_V\text{d}s < \infty \,\,\forall \,\,T >0\}.$$
One defines a metric $\rho$ on $\mathbb{B}$ through $$\rho(\omega_1,\omega_2) := \sum_{k=1}^{\infty}2^{-k}\bigg[\bigg(\int_{0}^{k}||\omega_2(s)-\omega_1(s)||_V \text{d}s + \underset{t \in [0,k]}{\text{sup}}||\omega_2(t)-\omega_1(t)||_{H}\bigg)\wedge 1\bigg].$$
Finally let $\pi_t:\mathbb{B} \to H$ denote the canonical projection at time $t \geq 0$, i.e. $\pi_t(y) := y_t$. Now I want to consider the Borel $\sigma$-algebra of the topology on $\mathbb{B}$, which is induced through $\rho$, denoted by $\mathcal{B}(\mathbb{B})$ as usual.
**My question: Is it true - and if yes: how does one show this - that ** $$\mathcal{B}(\mathbb{B}) = \sigma(\pi_t|t \geq 0)?$$
Clearly each $\pi_t$ is continuous and hence "$\supseteq$" is trivial. But the other inclusion does not seem to be clear to me. I am especially concerned, because the definition of the metric involves the $V$-norm, which is not equivalent to the norm on $H$, but might be much bigger. Hence - in general - the metric $\rho$ could be much bigger than just the weighted sup-norm (i.e. the metric we'd obtain by dropping the first summand for every $k$).
I would be very grateful for any hint on this! Thanks a lot!