Closed Graph Theorem for Banach spaces — If ${\displaystyle T:X\to Y}$ is an everywhere-defined linear operator between Banach spaces, then the following are equivalent:

1. ${\displaystyle T}$ is continuous.
2. ${\displaystyle T}$ is closed (that is, the graph of ${\displaystyle T}$ is closed in the product topology on ${\displaystyle X\times Y).}$
3. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x}$ in ${\displaystyle X}$ then ${\displaystyle T\left(x_{\bullet }\right):=\left(T\left(x_{i}\right)\right)_{i=1}^{\infty }\to T(x)}$ in ${\displaystyle Y.}$
4. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0}$ in ${\displaystyle X}$ then ${\displaystyle T\left(x_{\bullet }\right)\to 0}$ in ${\displaystyle Y.}$
5. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x}$ in ${\displaystyle X}$ and if ${\displaystyle T\left(x_{\bullet }\right)}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y,}$ then ${\displaystyle y=T(x).}$
6. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0}$ in ${\displaystyle X}$ and if ${\displaystyle T\left(x_{\bullet }\right)}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y,}$ then ${\displaystyle y=0.}$

Theorem — A linear operator from a barrelled space ${\displaystyle X}$ to a Fréchet space ${\displaystyle Y}$ is continuous if and only if its graph is closed.

Theorem — A linear map between two F-spaces is continuous if and only if its graph is closed. ^{[1] [2]}

Theorem — If ${\displaystyle T:X\to Y}$ is a linear map between two F-spaces, then the following are equivalent:

1. ${\displaystyle T}$ is continuous.
2. ${\displaystyle T}$ has a closed graph.
3. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to x}$ in ${\displaystyle X}$ and if ${\displaystyle T\left(x_{\bullet }\right):=\left(T\left(x_{i}\right)\right)_{i=1}^{\infty }}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y,}$ then ${\displaystyle y=T(x).}$ ^[3]
4. If ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i=1}^{\infty }\to 0}$ in ${\displaystyle X}$ and if ${\displaystyle T\left(x_{\bullet }\right)}$ converges in ${\displaystyle Y}$ to some ${\displaystyle y\in Y,}$ then ${\displaystyle y=0.}$

Closed Graph Theorem ^[4] — Also, a closed linear map from a locally convex ultrabarrelled space into a complete pseudometrizable TVS is continuous.

Closed Graph Theorem — A closed and bounded linear map from a locally convex infrabarreled space into a complete pseudometrizable locally convex space is continuous. ^[4]

Theorem ^[5] — Suppose that ${\displaystyle T:X\to Y}$ is a linear map whose graph is closed. If ${\displaystyle X}$ is an inductive limit of Baire TVSs and ${\displaystyle Y}$ is a webbed space then ${\displaystyle T}$ is continuous.

Closed Graph Theorem ^[4] — A closed surjective linear map from a complete pseudometrizable TVS onto a locally convex ultrabarrelled space is continuous.

Theorem ^[6] — Suppose that ${\displaystyle X}$ and ${\displaystyle Y}$ are two topological vector spaces (they need not be Hausdorff or locally convex) with the following property:

If ${\displaystyle G}$ is any closed subspace of ${\displaystyle X\times Y}$ and ${\displaystyle u}$ is any continuous map of ${\displaystyle G}$ onto ${\displaystyle X,}$ then ${\displaystyle u}$ is an open mapping.

Under this condition, if ${\displaystyle T:X\to Y}$ is a linear map whose graph is closed then ${\displaystyle T}$ is continuous.

Borel Graph Theorem — Let ${\displaystyle u:X\to Y}$ be linear map between two locally convex Hausdorff spaces ${\displaystyle X}$ and ${\displaystyle Y.}$ If ${\displaystyle X}$ is the inductive limit of an arbitrary family of Banach spaces, if ${\displaystyle Y}$ is a Souslin space, and if the graph of ${\displaystyle u}$ is a Borel set in ${\displaystyle X\times Y,}$ then ${\displaystyle u}$ is continuous. ^[7]

Generalized Borel Graph Theorem ^[8] — Let ${\displaystyle u:X\to Y}$ be a linear map between two locally convex Hausdorff spaces ${\displaystyle X}$ and ${\displaystyle Y.}$ If ${\displaystyle X}$ is the inductive limit of an arbitrary family of Banach spaces, if ${\displaystyle Y}$ is a K-analytic space, and if the graph of ${\displaystyle u}$ is closed in ${\displaystyle X\times Y,}$ then ${\displaystyle u}$ is continuous.