Showing that a set of operators is not closed

Question

I want to show that,

for $X$ a normed space, $T \in L(X;X)$,

the set $\{S \in L(X;X) : ST = TS\}$ is not closed in $L(X;X)$.

We might be able to construct a sequence $(S_n)$ whose the limit is not in $L(X;X )$. However, I am struggling to find a concrete example.

Answer 1

First, in what topology? A reasonable possibility is using the operator norm on $L(X,X)$. In the operator norm, the operator product $$L(X,X)\times L(X,X)\longrightarrow L(X,X)$$ $$(R,S)\longmapsto RS$$ is continuous and your set is obviously closed.