Let $X$ be a topological affine space. A Gaussian measure on $X$ is characterized by the property that its finite-dimensional projections are multivariate Gaussian distributions.

Is there a direct characterization of a Gaussian measure which does not rely on finite-dimensional projections? This definition is analogous to describing a duck as the animal whose shadows look like $2$-dimensional ducks. The definition is sufficient for doing analysis, but to me it misses the essence of what a Gaussian measure <i>is</i> as a mathematical object in and of itself. 

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Here is the precise definition of a Gaussian measure that I usually work with, which relies on the fact that Gaussians are entirely described by their covariance structure.

For $X$ a topological affine space as above, let $X^*$ denote its dual space of affine functionals. The dual space is a linear space, since there there is a natural zero functional $0 \in X^*$.

Let $K : X^* \to X$ be a continuous affine operator which is symmetric and non-negative-definite. i.e., $f'(Kf) = f(Kf')$ and $f(Kf) \ge 0$ for all $f, f' \in X^*$. Let $m_K := K(0)$ denote the image of the zero functional.

There is a unique Gaussian measure $P_K$ on $X$ with mean point $m_K \in X$ and covariance operator $K : X^* \to X$. That is, if $\pi : X \to \mathbb R^n$ denotes a finite-dimensional projection, then the push-forward measure $\pi_* P_K := P_K \circ \pi^{-1}$ is an $n$-dimensiona Gaussian distribution with mean vector $\pi(m_K) \in \mathbb R^n$ and covariance matrix $\pi K \pi^*$, where $\pi^* : (\mathbb R^n)^* \to K^*$ denotes the formal adjoint operator.

Furthermore, the <a href="http://en.wikipedia.org/wiki/Structure_theorem_for_Gaussian_measures">structure theorem for Gaussian measures</a> states that all Gaussian measures arise in this way. Consequently, we may parametrize the space of Gaussian measures by the space $\mathcal K(X)$ of symmetric, non-negative operators from $X^*$ to $X$.

This provides a weak answer to the question stated at the top of this post: yes, Gaussian measures can be directly characterized by their covariance structure. Consequently, here is the stronger form of my question:

 - Is there a geometric description of the space $\mathcal K(X)$ of Gaussian covariance operators?

For example, is the space $\mathcal K(X)$ an infinite-dimensional manifold? What is its symmetry group?