I am studying Gaussian measures on nuclear spaces, which for concreteness I take to be Schwartz space $\mathcal{S}'(\mathbb{R}^n)$. Just as in finite dimensions, these Gaussian measures can be uniquely characterized by their mean and covariance. However unlike in finite dimensions, the covariance acts on the base space $\mathcal{S}(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n)$. But given that the measure itself is defined on the dual space $\mathcal{S}'(\mathbb{R}^n)$, I cannot get a good intuition on why the covariance operator is not also defined on the dual space $\mathcal{S}'(\mathbb{R}^n) \times \mathcal{S}'(\mathbb{R}^n)$.
I think this is a technical detail that results from how the Gaussian measure is constructed (i.e. from finite dimensional subsapces) but I am not sure. Is there some better way to think of why the covaraince operator acts on $\mathcal{S}(\mathbb{R}^n) \times \mathcal{S}(\mathbb{R}^n)$ instead of $\mathcal{S}'(\mathbb{R}^n) \times \mathcal{S}'(\mathbb{R}^n)$?