probability - Why is Wiener measure a Gaussian measure?

This is silly and trivial so let me be really clear with my definitions and where exactly I got stuck.

The space $(C[0,T],\|\cdot\|_\infty,\sigma(\|\cdot\|_\infty),\gamma)$ is called classical Wiener space where $\gamma$ is Wiener measure.

I define Wiener measure as follows:

Wiener measure is the Kolmogorov extension of the finite dimensional distributions of the Wiener process.

And Gaussian measure on Banach space:

Let $\mathcal B$ be a Banach space with dual $\mathcal B^\ast$. A Borel measure $\gamma$ is called Gaussian iff for all $\ell\in \mathcal B^\ast$ the pushforward measure $\ell^\ast \gamma$ is a Gaussian measure on $\Bbb R$ where $\ell^\ast \gamma (A)=\gamma(\ell^{-1}(A))$ is the pushforward measure.

So I know the dual space of $C[0,T]$ is $\operatorname{RCA}([0,T])$ (by Riesz-Markov-Kakutani theorem) the space of all complex signed Radon measures with finite total variation. So we have to check that $\mu^\ast(\gamma)$ is a Gaussian measure on $\Bbb R$ for all $\mu\in \operatorname{RCA}([0,T])$.

If $\mu$ is a finite linear combination of $\delta$s we are fine as $\delta_{t}^{-1}(A)$ are all the paths that pass through the set $A$ at time $t\in [0,T]$. Then $\gamma$ of this is Gaussian by definition. Linearity is not too bad after this.

Then we have to extend to all measures and I am not sure how to do this. I know $\delta$s should be dense in $\operatorname{RCA}([0,T])$ and Wiener measure is the Kolmogorov extension of the finite dimensions but I'm not sure exactly what the argument should be.