# Abstract Wiener Space for $\ell^2(\mathbb{R})$

Let $$\ell^2(\mathbb{R})$$ denote the space of square-summable real-valued sequences equipped with the inner product

$$\langle x, y \rangle = \sum_{n = 1}^{\infty} x_n y_n$$

Let $$\nu$$ denote its canonical cylinder measure i.e. a cylinder measure defined on the weak sigma algebra whose Fourier transform has the form

$$\exp( - \frac{1}{2} \langle x, x \rangle_{\ell^2}) ~~.$$

Does there exist a measurable norm associated to $$\ell^2$$ and $$\nu$$?

If so, is there a nice description of its associated abstract Wiener space?

If not, is there a proof that there is not?

I do know that $$\ell^2$$ is the Cameron-Martin space of $$\mathbb{R}^{\infty}$$ with the product topology and the distribution of an iid sequence of random variables $$(X_n)_{n \in \mathbb{N}}$$ with $$X_1 \sim \mathcal{N}(0,1)$$, and also that $$\ell^2$$ lies dense in that space w.r.t. the topology of point-wise convergence.

So it "should be" $$\mathbb{R}^{\infty}$$, but it is not since $$\mathbb{R}^{\infty}$$ is only separable Frechet and not Banach.

Sure, there are many such norms (they are highly non-unique). Indeed, take any other infinite-dimensional abstract Wiener space $$(W, H, \mu)$$. Since $$H$$ and $$\ell^2$$ are both infinite-dimensional separable Hilbert spaces, there is an isometric isomorphism $$T : \ell^2 \to H$$. Now since $$\|\cdot\|_W$$ is a measurable norm on $$H$$, it follows that $$\|T \cdot\|_W$$ is a measurable norm on $$\ell^2$$.
If you want the norm to be an inner product, let $$A$$ be any injective Hilbert-Schmidt operator on $$\ell^2$$. It is shown in Proposition 4.59 of these notes that $$\langle Ax, Ay\rangle$$ defines an inner product on $$\ell^2$$ which induces a measurable norm. (The assumption that $$A$$ should be injective is missing from the notes, but is necessary; otherwise we only get a measurable seminorm.)
For a specific example, fix a sequence of real numbers $$a_n$$ with $$\sum a_n^2 < \infty$$ (such as $$a_n = 1/n$$) and apply the proposition to the operator defined by $$(Ax)_n = a_n x_n$$. The resulting inner product is $$\langle x,y \rangle_a = \sum a_n^2 x_n y_n$$, and the completion of $$\ell^2$$ under this inner product is simply the weighted $$\ell^2$$ space $$\ell^2_a$$ of sequences $$x$$ satisfying $$\sum a_n^2 x_n^2 < \infty$$. It's easy to construct the associated Gaussian measure on $$\ell^2_a$$; let $$\xi_n$$ be an iid sequence of standard normal random variables on some probability space $$\Omega$$. If $$e_n$$ is the standard basis in $$\ell^2$$, you can easily show that $$\sum \xi_n e_n$$ converges in $$L^2(\Omega; \ell^2_a)$$, and $$\mu$$ is the law of its limit.