White noise is a Borel Gaussian law on the negative Sobolev space

Write $w_l=a_l^{-1}(\log a_l{)}^{-2\delta }$. In the wavelet case, the trace sum is a finite coarsest term plus ${\sum }_{l\ge 1}{2}^l{w}_l=(\log 2{)}^{-2\delta }{\sum }_{l\ge 1}{l}^{-2\delta }$. On each block $2^m\le l<2^{m+1}$, the sum of $l^{-r}$ is between $2^m{2}^{-r(m+1)}$ and $2^m{2}^{-rm}$; hence these blocks form a convergent geometric series when $r>1$, while for $r\le 1$ their contributions are bounded below by a fixed positive constant and the series diverges. For trigonometric coefficients with bounded multiplicities, the block $2^m\le \mathrm{\mid }l\mathrm{\mid }<2^{m+1}$ contributes between constant multiples of $m^{-2\delta }$ to ${\sum }_{l\ge 2}[l(\log l{)}^{2\delta }{]}^{-1}$. The same comparison therefore gives finiteness exactly when $\delta >1\mathrm{/}2$. Thus the trace $T_{\delta }:={\sum }_{l,k}{w}_l$ is finite in either case.

Let $S_m$ be the sum of $w_l{Z}_{lk}^2$ over the first $m$ coordinates in the fixed countable enumeration. These nonnegative partial sums increase to $S={\sum }_{l,k}{w}_l{Z}_{lk}^2$, and [F3] gives $\mathbb{E}S={\lim }_m\mathbb{E}{S}_m={\sum }_{l,k}{w}_l=T_{\delta }$. The mean and second moment of each coordinate are $0$ and $1$ by [F5]. If $S=\mathrm{\infty }$ on a set of probability $p>0$, then $\mathbb{E}S\ge Rp$ for every $R>0$, impossible; hence $S<\mathrm{\infty }$ almost surely.

On that probability-one event, the finite-coordinate sums of $W$ are Cauchy in $H(\delta )$ because the squared norm of every tail is the corresponding tail of the convergent series $S$. Completeness gives a limit in $H(\delta )$. The Cauchy event is measurable, and the partial sums are measurable; their pointwise limit on that event is measurable as an $H(\delta )$-valued map because the space is separable. Setting it to zero on the measurable complement gives a Borel random element.

Let $e_j$ be the unit vector of the weighted sequence Hilbert space supported on coordinate $j$ and let $b_j=L(e_j)$ for a continuous real linear functional $L$. For each finite set $F$, boundedness applied to ${\sum }_{j\in F}{b}_j{e}_j$ gives ${\sum }_{j\in F}{b}_j^2\le \mathrm{\parallel }L{\mathrm{\parallel }}^2$, so ${\sum }_j{b}_j^2<\mathrm{\infty }$. Since the weights $w_j$ are bounded, $v:={\sum }_j{w}_j{b}_j^2<\mathrm{\infty }$. The finite-coordinate projections of $W$ converge almost surely in $H(\delta )$ by step 3.1, so continuity gives $L(W)={\lim }_m{\sum }_{j\le m}\sqrt{w_j}{b}_j{Z}_j$ almost surely. By [F5] this limit has exactly the density normal law $N(0,v)$. Thus $W$ is centered Gaussian. For a continuous coefficient pairing $L(c)={\sum }_j{d}_j{c}_j$, one has $b_j=d_j\mathrm{/}\sqrt{w_j}$, giving $v={\sum }_j{d}_j^2$.

Enumerate the infinite basis as $({\psi }_j{)}_{j\ge 1}$, write its white-noise coordinates as $Z_j$, and let $A_j=\{\mathrm{\mid }{Z}_j\mathrm{\mid }\ge 1\}$. The displayed normal density is positive on $[1,2]$, so $p:=\mathbb{P}(A_j)>0$; independence gives $\mathbb{P}({\bigcap }_{j=m}^{m+q}{A}_j^c)=(1-p{)}^{q+1}$. By [F4], $\mathbb{P}({\bigcap }_{j\ge m}{A}_j^c)=0$ for each $m$, so the event that only finitely many $A_j$ occur, a countable union of these intersections, has probability zero: disjointify that union by replacing its $m$th event $E_m$ with $E_m\setminus {\bigcup }_{i<m}{E}_i$, and use countable additivity and monotonicity to sum zero probabilities. Therefore ${\sum }_{j\le n}{Z}_j^2$ is unbounded almost surely. If the same $H(\delta )$ element were represented by an $L^2$ function, the coefficient embedding in [F1] would identify its $L^2$ coefficients with $Z_j$. The finite Pythagorean identity, applied to the projection onto ${\psi }_1,\dots ,{\psi }_n$, would then give ${\sum }_{j\le n}{Z}_j^2\le \mathrm{\parallel }W{\mathrm{\parallel }}_2^2$ for every $n$, a contradiction.