Vtyjkyuk

I am reading Nualart's book The Malliavin Calculus and Related Topics and there is some issues that I am stuck wtih.

Let $H$ be a real separable Hilbert Space with. A stochastic process $W=W(h);;hin H$ defined in a complete probability space $(Omega,mathcalF,P)$ is an isonormal Gaussian process if $W$ is a centered Gaussian family of random variables such that $E(W(h)W(g))=langle h,grangle$ for all $g,hin H.$

The map $hmapsto W(h)$ is linear. Now Nualart said

1. The mapping $hto W(h)$ provides a linear isometry of $H$ onoto a closed subspace of $L^2$ that will denote by $mathcalH_1.$

Question $1$: Why the map provides a linear isometry ?

Denote, now, by $mathcalG$ the $sigma-$alebra generated by the random variables $W(h);; hin H.$

2. The set $exp(W(h)),hin H$ form a total subset of $L^2(Omega,mathcalG,P)$

The proof goes as follow, we take $Xin L^2$ such that $E(Xe^W(h))=0$ for all $hin H.$

By linearity of $hto W(h)$ we have $$Ebig( Xexp(sum_i=1^n t_iW(h_i))big)=0.quad (3)$$

This equation says that Laplace transform of the signed measure $Ebig(Xmathbb1_B(W(h_1),W(h_2),ldots,W(h_n))big)$ is identically zero on $BbbR^n.$

Then $E(Xmathbb1_G)=0$ for all $GinmathcalG$ so that $X=0.$

Quesstion $2$: I don't understand why linearity of $hto W(h)$ gives us equation $(3)$ and cannot write down that is the Laplace transform of of the signed measure $Ebig(Xmathbb1_B(W(h_1),W(h_2),ldots,W(h_n))big).$

First question: $|W(g)|^2=langle W(g), W(g) rangle =langle g, g rangle =|g|^2$ so $|W(g_1)-W(g_2)|=|W(g_1-g_2)|=|g_1-g_2||$. This proves that $g to W(g)$ is an isometry. To prove (3) all you have to do is write $sum_i=1^n t_iW(h_i)$ as $W(h)$ with $h =sum_i=1^n t_i h_i$. For the last part you need the following fact: suppose $mu(B)=EXI_B(Z)$ for Borel sets $B$ in $mathbb R^n$ where $Z$ is a random vector with $n$ components. Then, for any non-negative measurable function $f$ we have $int f, dmu =EXf(Z)$. To prove this last equation you just have to note that it holds when $f$ is an indiactor function (by definition), hence for all simple functions $f$, hence for all non-negative measurable functions $f$.