Let $X$ be a locally convex topological linear space, and let $\mathbb P$ be a probability measure on $X$. Suppose that $\operatorname{var}(\varphi) < \infty$ for all continuous linear functionals $\varphi \in X^\*$. By Theorem 3.ii of [[Vakhania & Tarieladze 1978]](http://epubs.siam.org/doi/pdf/10.1137/1123001), the embedding $i : X^\* \to L^2(X,\mathbb P)$ is continuous.


Let $C \subseteq L^2(X,\mathbb P)$ denote the subspace of almost-surely constant functions. 


Is the direct sum $iX^\* \oplus C$ dense in $L^2(X,\mathbb P)$?



Density of linear functionals in $L^2$
As requested, I'll elaborate here on the Gaussian case.


Suppose for instance that $X$ is a separable Banach space, and $\mathbb{P}$ is a Gaussian measure. Then each $f \in X^\*$, when considered as a random variable on $(X, \mathbb{P})$ has a Gaussian distribution. Let $H$ denote the $L^2$-closure of $i X^\*$. An $L^2$ limit of Gaussian random variables remains Gaussian, so every element of $H$ (and hence also $C \oplus H$) is Gaussian. But $L^2(X,\mathbb{P})$ clearly contains lots of non-Gaussian random variables (even when $X = \mathbb{R}^n$); as just one example, those of the form $f^2$ have a $\chi^2$ distribution.


However, since each $f \in X^\*$ is Gaussian and hence has all moments, then for any polynomial $p$ in any number of variables $n$, we have that $p(f\_1, \dots, f\_n) \in L^2$. Then it can be shown by a Stone-Weierstrass type argument that the set of all such polynomials is indeed dense in $L^2$ (it's conveniently done with a functional version of the monotone class or $\pi$-$\lambda$ theorem). A key point is that because of the separability of $X$, one can show that $X^\*$ generates the Borel $\sigma$-algebra of $X$. (Note that it is not necessary that $X^\*$ be separable.)


This can be done in a more orderly way, too. We can find a countable sequence $e\_i \in X^\*$ which are $L^2$-orthonormal and span an $L^2$-dense subspace of $X^\*$. Let $H\_m(s)$ be the $m$th Hermite polynomial (e.g. $H\_0(s) = 1$, $H\_1(s) = s$, $H\_2(s) = s^2 - 1$, etc) and consider the polynomials of the form
$$F\_{a\_1, \dots, a\_k}(x) = H\_{a\_1}(e\_1(x)) \cdots H\_{a\_k}(e\_k(x)).$$
These functions form an orthogonal basis of $L^2$. Moreover, if we let $\mathcal{H}\_n$ be the closed linear span of $\{F\_{a\_1, \dots, a\_k} : a\_1 + \dots + a\_k = n\}$, then we get an orthogonal decomposition $L^2 = \bigoplus\_{n=0}^\infty \mathcal{H}\_n$. This is the so-called Wiener chaos decomposition. Note that $\mathcal{H}\_n$ is naturally isomorphic to the $n$-fold symmetric tensor of $H$ with itself, and so we get an isomorphism of $L^2$ with the bosonic Fock space over $H$.


In particular, $\mathcal{H}\_0 = C$ and $\mathcal{H}\_1 = H$. So you were on the right track, but you stopped too soon :)


I wrote a little bit about this in [these lecture notes](http://www.unco.edu/NHS/mathsci/facstaff/Eldredge/7770/7770-lecture-notes.pdf), which also has a few references.


