In finite dimensions, a Gaussian distribution has full support if and only if its covariance matrix is positive definite. I am now wondering whether this holds true also for the infinite-dimensional case, i.e. for Gaussian processes.
More precisely, let $T$ be a compact metric space (possibly separable and complete if necessary) and $X$ a Gaussian process in $C(T,\mathbb{R})$ with continuous mean function $m\colon T\to \mathbb{R}$ and positive definite covariance function $K\colon T\times T\to \mathbb{R}$. Then the distribution of $X$ is a probability measure on the Borel sets of $C(T,\mathbb{R})$. Does this measure have full support? If not, are there some further assumptions on $T$, $m$, and $K$ which would imply this?
By support of a measure I mean the complement of the largest open set with measure zero.
If a similar statement does hold true, I would be grateful for a proof or a source where I can find one.