I am trying to get my head around this question and was reading (1) which states the same a little bit more general:

Let $X$ be a separable Banach space and $X^*$ the dual space. The mean value $m \in X$ and covariance $C\colon X^*\to X$ are defined by \begin{align} x^*(m) &= \int_X x^*(x) \mu(dx),\\ y^*(Cx^*) &= \int_X x^*(x)y^*(x) \mu(dx) - x^*(m)y^*(m) \end{align} $x^*,y*\in X^*$ and measure $\mu$.

Moreover

A measure $\mu$ is called Gaussian if all the functionals $x^*\in X^*$, regarded as random variables on $(X,\mu)$ are Gaussian random variables.

I considered $X=\mathbb{R}^2$ since this is easier to visualize and I can check the results with my experience of $\mathbb{R}^2$ random variables.

I generated samples $Z$ from a Gaussian distribution with mean $m = (1,1)^T$ and covariance $C = \bigl(\begin{smallmatrix} 1&0.5 \\ 0.5&0.4 \end{smallmatrix} \bigr)$.

Choosing a $x^* = (2,3)$ (or any other value) then $x^*(Z)$ seems to be Gaussian with $x^*(m) = (2,3)\cdot(1,1)^T = 5$ and $x^*(Cx^*) = (2,3)\cdot (C(2,3)\cdot (2,3)) = 13.6$ as in the histogram:

and everything seems to be fine.

My question now is, can I somehow access the density of the (multivariate Gaussian) distribution **at** $x^*$ if $X$ is a Banach space, to get the value of
\begin{align}
f(x^*) = \frac{1}{\sqrt{2\pi^2|C|}} \exp\big(-0.5(x^*-m)^TC^{-1}(x^*-m) \big)
\end{align}
It would be great to get an example assuming that $X$ and the measure are sufficiently "nice" (eg. the measure is not degenerate etc, X is separable or a Hilbert space, etc.).

**Edit:** As Nate Eldredge pointed out there is no Lebesgue measure. Let me give a possible application: Failure detection on an engine. There is some sensor delivering a curve (function) which carries information about the engine properties. So each test of an enigne delivers a curve (function) and therefore we model $X$ to be a separable Hilbert space of square-integrable functions. A malfunction is now detection if a test yields a curve outside two standard deviations.

(1) Rudnìcki, Ryszard. "Gaussian measure-preserving linear transformations." Univ. Iagel. Acta Math 30 (1993): 105-112.