prove a subset of squence space $l^p$ closed in strong topology

Question

Let $l^p$ be the space of $p$-summable sequences. von Neumann constructed a subset of $l^p$ space $$S=\{X_{mn}: m,n≥1\}$$ where $X_{mn}\in l^p$ are defined by $X_{mn}(m)=1, X_{mn}(n)=m$ and $X_{mn}(k)=0$ otherwise.

I am asked to show that this $S$ is closed in the strong topology. I tried to show the complement is open by trying to construct a contradiction, but no success. Could anyone help me ? Thanks in advance.

Answer 1

Hints:

1. Prove the following general claim. The set $S$ in metric space $M$ with the property $$ \exists C>0\quad \forall x'\in S\quad\forall x''\in S\quad (x'\neq x''\implies d(x',x')>C)\tag{1} $$ is always closed.

2. Prove that the set $S$ in your problem satisfy this condition.