I'll begin this question with the finite-dimensional case, as a warmup.

Let me say a continuous path $\omega : [0,1] \to \mathbb{R}^d$ is
**hyperplanar** if there exists a nonzero $x \in \mathbb{R}^d$ such
that $\omega(t) \cdot x = 0$ for all $t \in [0,1]$.

It is not hard to show:

Proposition.Let $B_t$ be a standard Brownian motion in $\mathbb{R}^d$. Almost surely, $(B_t : 0 \le t \le 1)$ is not hyperplanar.

To avoid possible confusion, let me emphasize that the hyperplane is allowed to be random. That is, I am claiming: $$\mathbb{P}(\exists x \in \mathbb{R}^d\\, \forall t \in [0,1] : B_t \cdot x = 0) = 0$$ which should not be confused with the weaker statement $$\forall x \in \mathbb{R}^d :\mathbb{P}(\forall t \in [0,1] : B_t \cdot x = 0) = 0.$$

One possible proof of the proposition is to choose $0 < t_1 < \dots < t_d < 1$, and show by induction that, almost surely, $B_{t_1}, \dots, B_{t_d}$ are linearly independent in $\mathbb{R}^d$. (By the induction hypothesis, $B_{t_1}, \dots, B_{t_{k-1}}$ span a $k-1$-dimensional subspace of $\mathbb{R}^d$; by the Markov property and the absolute continuity of the Gaussian distribution, $B_{t_k}$ is almost surely not in this subspace.)

I am interested in the analogous statement for infinite dimensions.
Let $W$ be a real separable Banach space, and say a continuous path $\omega :
[0,1] \to W$ is **hyperplanar** if there exists a nonzero continuous
linear functional $f \in W^*$ such that $f(\omega(t)) = 0$ for all $t
\in [0,1]$. Let $\mu$ be a non-degenerate Gaussian measure on $W$
(so that $(W,\mu)$ is an abstract Wiener space), and let $B_t$ be a standard Brownian motion in $W$. That is, the process
$B_t$ has continuous sample paths and independent increments, starts
at 0, and the increments are distributed such that $(t-s)^{-1/2}(B_t -
B_s) \sim \mu$.

For infinite-dimensional $W$, what is the probability that $(B_t : 0 \le t \le 1)$ is hyperplanar?

As a start, I can show that the set of hyperplanar paths is analytic in $C([0,1], W)$, and hence universally measurable, so the question actually makes sense. We can also use a scaling argument and the Blumenthal zero-one law to see that the probability must be either 0 or 1.

I am not sure which way my intuition points here. On the one hand, we expect a Brownian motion to be pretty irregular and unconstrained, suggesting the answer is 0 as in finite dimensions. On the other hand, an infinite-dimensional space has a lot of hyperplanes. In principle, the answer could depend on the abstract Wiener space $(W,\mu)$.

Thanks!