Smooth functionals extend continuously to the white-noise topology

Since $s>1\mathrm{/}2$, the level factor $a_l^{1-2s}(\log a_l{)}^{2\delta }$ is bounded: it tends to zero as $a_l\to \mathrm{\infty }$, because a positive power of $a_l$ dominates every fixed power of $\log a_l$, and the finitely many coarsest levels have finite weights. Thus $w_l^{-1}\le C_{s,\delta }^2{a}_l^{2s}$ after increasing a finite constant.

Weighted Cauchy–Schwarz gives ${\sum }_{l,k}\mathrm{\mid }{x}_{lk}{g}_{lk}\mathrm{\mid }\le ({\sum }_{l,k}{w}_l\mathrm{\mid }{x}_{lk}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}({\sum }_{l,k}{w}_l^{-1}\mathrm{\mid }{g}_{lk}{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}\le C_{s,\delta }\mathrm{\parallel }x{\mathrm{\parallel }}_{H(\delta )}\mathrm{\parallel }g{\mathrm{\parallel }}_{H_2^s}$. The same bound applied to $x-y$ proves continuity and absolute convergence of the series.

If $x$ is the coefficient sequence of $f\in L^2$, its finite orthonormal projections converge to $f$ in $L^2$ because $({\psi }_{lk})$ is a complete orthonormal basis. Their inner products with $g$ are the finite partial sums of the displayed pairing, and converge to $⟨f,g{⟩}_{L^2}$ by continuity of the inner product. Step 2.1 identifies the same limit with $L_g(x)$.