In Theorem 3.1 of this paper, the following result is formulated:
Suppose $f:S_1 \to \mathbb{R}$ is a continuous function where $S_1$ is a compact subset of $S(\mathcal{V}^p(J,E))$. Then for any $\epsilon > 0$, there exists a linear functional $L \in T((E))^*$ such that for every $a \in S_1$, $$ |f(a)-L(a)| \leq \epsilon. $$
Here $\mathcal{V}^p(J,E)$ denotes the space of paths of finite $p$-variation from an interval $J$ to a Banach space $E$ (assume finite dimensional), and $S$ is the signature map defined as a collection of iterated Young integrals.
My question is, what is the topology on $S(\mathcal{V}^p(J,E))$? The proof uses the Stone-Weierstrass theorem, and I think that even requires $S_1$ to be a metric space, but since $S(\mathcal{V}^p(J,E))$ is a subset of the full tensor algebra $T((E))$ and is sort of infinite dimensional (even assuming $E$ is finite dimensional), I don't see what topology or metric it should have.
Or maybe this theorem should be interpreted as: whatever suitable topology we give this space, we can approximate continuous, real-valued functions by linear functionals?
Let $E=\mathbb{R}^d$. Each $E^{\otimes n}= (\mathbb{R}^d)^{\otimes n}$ can be endowed with the usual distance, e.g. when $a^2,b^2 \in (\mathbb{R}^d)^{\otimes 2} = \mathbb{R}^{d\times d}$, we can define $$ d_2(a^2,b^2) = \max_{1\le i,j \le d} |a^2_{i,j}-b^2_{i,j}|. $$ Then for $a = (1,a^1,a^2,\ldots),b=(1,b^1,b^2,\ldots) \in T((E)) = \bigoplus_{n=1}^{\infty} E^{\otimes n}, $ $$ d(a,b) = \sum_{n=1}^{\infty}\frac{d_n(a^n,b^n)}{2^n}$$ defines a natural distance between $a$ and $b$.
I'm not sure though if this is the topology the authors of the paper had in mind.