# Closed map $T:X \to Y$ has closed graph?

Let $T:X\to Y$ be a linear operator between two normed vector spaces. My question is:

If $T$ is a closed map (sends closed sets to closed), then is the graph of $T$ a closed set of $X \times Y$?

It seems to be the case according to this question, even if the converse is not true. I haven't found a counter-example.

• @TsemoAristide : thank you. 1. Ah yes, for instance here. But if $T$ is not assumed to be continuous, but only linear? $\tag*{}$ 2. For the converse, don't we need $X$ and $Y$ to be Banach spaces? For me, the closed graph theorem tells us that the graph of $T$ is closed iff $T$ is continuous. In particular, I would get that $T$ is closed? – Alphonse May 29 '16 at 13:22