Let $X$ be a ***separable topological vector space*** with size (cardinal number) no larger than $\mathfrak{c}$. Does there exist any sequence of finite rank linear maps $\phi\_n:X\to X$ pointwise converging to the identity mapping $id:X\to X$? 



pointwise convergence to the identity
Just to chat, an easier counterexample is $X:=L^p(\mathbb{R})$ for $0\le p<1$, a complete metric separable TVS. The identity map can't be approximated by finite rank continuous linear operators, for the simple reason that there aren't any. The only convex open set of $X$ is $X$ itself. As a consequence, there aren't any non-zero continuous linear forms on $X$, nor any non-zero finite rank continuous linear operators. 


