Closed Graph Theorem

Closed Graph Theorem

An (everywhere-defined) linear operator between Banach spaces is bounded iff its graph is closed.

We prove a slightly more general version:

Closed Graph Theorem

Let $X$ and $Y$ be Banach spaces and $T : \dom{T} \to Y$ a linear operator with domain $\dom{T}$ closed in $X$. Then $T$ is bounded if and only if its graph $\graph{T}$ is closed.

Proof: $\square\enspace$