Let $\{e_i\}_{i\in N}$ be an orthonormal basis in $L^2(R^m)$.
Take an arbitrary $\varphi_{1}\in {\rm L}^{s}(R^{d};{\rm L}^2(R^m))$, $s>1$.
Does it hold \begin{equation} \lim\limits_{M\to \infty}\| \varphi_1-\sum\limits_{i=1}^M c_{i}\, e_i\|_{L^{s}(R^d;L^2(R^m))}=0, \end{equation} where $c_{i}(x)=\int_{R^m}\varphi_1(p,x)e_i(p)dp$ and $\left(\sum_i c_i^2\right)^{1/2}$ belongs to ${\rm L}^{s}(R^d)$.
Thanks a lot!
Another way to phrase the question: let $P_k:L^2(\mathbb R^m)\to L^2(\mathbb R^m)$ be the projection on the subspace spanned by the first $k$ basis vectors. Is it true that $P_k\circ \varphi\to \varphi$ in $L^s$?
Each map $\varphi\to P_k\circ \varphi$ is a contraction on $L^s$. Therefore, it suffices to consider compactly supported continuous $\varphi$; the general case follows by approximation.
The range of $\varphi$ is a compact subset of $L^2(\mathbb R^m)$. We have $P_k\to I$ uniformly on this set. Therefore, $P_k\circ \varphi\to \varphi$ uniformly, and a fortiori in $L^s$.
