Suppose $V_i$ are subspaces of $B(H_i,K_i)$. Is it true that $\prod_i V_i$ is also a sub space of $B(H,K)$ for some appropriate $H$ and $K$?
I think we need to take $H$ and $K$ to be direct product of $H_i$ and $K_i$ respectively but I am not sure. Can somebody help me?