I define a Gaussian measure on a separable Banach space $(\mathcal{F},\|\cdot\|)$ as in van der Vaart and van Zanten (2008): A Borel measurable random element $W$ with values in a separable Banach space $(\mathcal{F},\|\cdot\|)$ is called Gaussian if the random variable $\varphi W$ is normally distributed for any element $\varphi$ of the dual space $\mathcal{F}^{*}$ of $\mathcal{F}$, and it is called zero-mean if the mean of every such variable $\varphi W$ is zero. And let $P=L(W)$ denote the zero-mean Gaussian measure.
Now I restrict attention to $\mathcal{F}$ consisting of real functions with domains on a set $\mathcal{X}$ (may or may not be compact), and $\mathcal{F}$ is dense in $C(\mathcal{X})$, the space of all continuous functions from $\mathcal{X}$ to $\mathbb{R}$, with respect to the uniform norm $\|\cdot\|_{\infty}$ defined as $\|f\|_{\infty}=\sup_{x\in\mathcal{X}}f(x)$. Can I say the unit ball with respect to the norm $\|\cdot\|$, denoted as $B(f, \|\cdot\|)$, has $P\{B(f, \|\cdot\|)\}>0$ for $\forall f\in \mathcal{F}$? And can I say the unit ball with respect to the uniform norm $\|\cdot\|_{\infty}$, has $P\{B(f, \|\cdot\|_{\infty})\}>0$ for $\forall f\in \mathcal{F}$?