Unbounded and close.

Question

Let $A$ a unbounded operator $ B \rightarrow B$ between $\mathbb{C}$ Banach space.

I would like to know if the following fact is true or false.

If the domain of $A$, $D(A)$, is dense in $B$ then $A$ is closed ?

Thanks and regards.

Answer 1

If it is true whenever $D(A)$ is dense it should also be true whenever $D(A)=B$. But when $D(A)=B$ the operator is closed iff it is bounded by Closed Graph Theorem. So any unbounded operator with domain $B$ is a counter-example to your statement.