Let $H$ be an (infinite dimensional) Hilbert space and $U$ another Hilbert space such that the embedding $H \rightarrow U$ is Hilbert-Schmidt. Now set an hilbertian basis $(e_k)$ of $H$, then $\sum \|e_k\|_{U}^2 < + \infty$. Let $(w^n_t)_{n}$ be a family of independant brownian motions (in one dimension).
One can show that $W_t:=\displaystyle \sum_{n} w^n_t e_n$ is convergent in $L^2(\Omega, U)$ (where $\Omega$ is the probability space on which the $w^n_t$ are defined).
Let $(\varepsilon _k)_k$ be another Hilbert basis of $H$. Then $\displaystyle \overline{w}^k_t := \sum_n w^n_t (e_n,\varepsilon_k)_H$ is well defined on $L^2(\Omega,\mathbb{R})$ and form a family of independant brownian motions.
I want to prove that one have $\displaystyle W_t = \sum_k \overline{w}_t^k \varepsilon _k$, almost surely on $U$.
What I've done so far. A formal computation that gives the result : $$\sum_k \overline{w}_t^k \varepsilon _k = \sum_k \sum_n w^n(e_n,\varepsilon_k)_H \varepsilon _k = \sum_n \sum_k w^n(e_n,\varepsilon_k)_H \varepsilon _k =W(t) $$