Agreement of two topologies on a linear space

Question

I'm dealing with the formalism of an abstract Wiener space, and I'm not sure if two relevant topologies coincide.

Let $X$ be a topological vector space, and let $X^*$ be its dual space of continuous linear functionals. (if $X$ is not locally convex then $X^*$ is trivial, but that's fine for these purposes). Let $K : X^* → X$ be a covariance operator, that is, a non-negative-definite and symmetric operator.

It is not hard to see that $K$ generates an inner product on $X^*$, defined by $\langle \psi | \varphi \rangle_K := \psi[K\varphi]$. Let $H_K$ denote the Hilbert-space completion of $X^*$ with respect to this inner product. (that is, $H_K \cong \overline{X^*/\ker K}$. It is not hard to see that there exist maps $i^* : X^* \to H_K$ and $i : H_K \to X$ so that $K = i \circ i^*$. That is, the following diagram commutes: $$\begin{array}[ccccc] ~X^* && \longrightarrow && X \\ & \searrow && \nearrow & \\ && H_K && \end{array}$$

The image $A_K := iH_K$ is called the Cameron-Martin space corresponding to the covariance operator $K$. I refer to the space $C_K := \overline{KX^*} = \overline{A_K}$. as the "outer core" for the operator. This closed linear subspace is of great interest in probability theory. If $\mathbb P$ is any Radon probability measure on $X$ with covariance operator $K$, then $\operatorname{supp}(\mathbb P) \subseteq C_K$; if $\mathbb P$ is a Gaussian measure, then $\operatorname{supp}(\mathbb P) = C_K$.

Here are my questions:

1. is the space $C_K$ equipped with a norm, which extends the norm on the Cameron-Martin subspace $A_K$?

2. if so, is this norm complete? (i.e., is $C_K$ naturally a Banach space?)

3. does this Banach topology agree with the subspace topology on $C_K \subseteq X$?

Let me know if this doesn't make sense, and I'll be happy to expand on any point.

Answer 1

What do you think about this rather trivial example: Let $X=c_0$ be the Banach space of (real) null sequences, $X^*=\ell_1$ its dual (where $y=(y_n)_{n} \in \ell_1$ is considered as the functional $(x_n)_n \mapsto \sum_n x_n y_n$) and $K:X^* \to X$ the inclusion. Then $A_K=\ell_2$ is dense in $c_0$ so that $C_K=X$. In this situation 3. is certainly false.

In general, if $(X,\mathscr T)$ is a Frechet space (by the way, don't you need some completeness of $X$ in order to have $i: H_K \to X$ well defined?), the closed graph theorem is an essential obstacle against your wishes: The Banach topology on $C_K$ is either equal to the Frechet space topology (so that $A_K$ would be closed in $X$ which hardly happens) or it must be so strange that you can not apply the closed graph theorem (in particular, there is no Hausdorff topology on $X$ which is coarser than the Frechet topology of $X$ and the Banach topology of $C_K$). I doubt that such a strange topology on $C_K$ would be of any use.