Closed graph theorem

Statement

Let $X$ and $Y$ be Banach spaces over the same scalar field $\mathbb R$ or $\mathbb C$ , and let $T:X\to Y$ be linear and defined on all of $X$ . If $T$ is a closed linear operator, then $T$ is bounded. Explicitly, closedness means that

x_n\to x\text{ in }X,\quad Tx_n\to y\text{ in }Y \quad\Longrightarrow\quad Tx=y.

Thus a topological condition on the graph forces continuity, provided both spaces are complete and the domain of $T$ is all of $X$ .

Relation to the open mapping theorem

Give the graph $\Gamma(T)\subseteq X\times Y$ the product norm. Closedness makes $\Gamma(T)$ a Banach space. The first-coordinate projection $\pi_X:\Gamma(T)\to X$ is a bounded linear bijection, so the open mapping theorem makes its inverse bounded. Composing that inverse with the second-coordinate projection shows that $T$ is bounded.

Domain-sensitive scope

The full-domain hypothesis is indispensable. On $\ell^2$ , let

D(T)=\{x=(x_n): (nx_n)\in\ell^2\},\qquad Tx=(nx_n).

This operator is closed and unbounded, but $D(T)$ is a proper dense subspace of $\ell^2$ . Such examples are the normal setting for unbounded differential and spectral operators; their domains are part of their definitions. Conversely, every bounded operator between normed spaces has a closed graph when the codomain is Hausdorff.

References

John B. Conway, A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics 96, Springer, 1990. Springer DOI record. Relevant: Chapter VI, “Linear Operators on a Banach Space.”
Walter Rudin, Functional Analysis, 2nd ed., McGraw–Hill, 1991. WorldCat record. Relevant: Chapter 2, the closed graph theorem.