So I am stuck at this situation . Suppose $m:B_2(H_1)\times B_2(H_2)\to \mathbb C$ be bilinear form given by $m(S,T)=\left<T,\phi(S)\right>$, where $\phi: B_2(H_1)\to B_2(H_2)$ be a bounded linear map and $B_2(H)$ denotes the space of Hilbert Schmidt operators on $H$ with inner product $\left<S,T\right>=Tr(T^*S)$. Let $\{S_i\}$,$\{\tilde S_j\}$ and$\{T_i\}$,$\{\tilde T_j\}$ be two pair of sequence in closed unit ball of $B_2(H_1)$ and $B_2(H_2)$ respectively such that the limit $$\lim_jm(S_i\tilde S_j,\phi(T_i\tilde T_j))<\infty$$ for each $i$

Which means, for each $i$ $$\lim_j \sum_{n\in I}\left<T_i\tilde T_je_n,\phi(S_i\tilde S_j)e_n\right><\infty$$ $\{e_n\}$ is orthonormal basis for $H_2$. Under what conditions can I take limit inside the summation ? I feel like using some variant of dominated convergence theorem with respect to counting measure but cannot figure out how. Are the limit and summation even interchangeable here?