--- title: Closed Graph Theorem parent: The Fundamental Four grand_parent: Functional Analysis Basics nav_order: 4 --- # {{ page.title }} {% theorem * Closed Graph Theorem %} An (everywhere-defined) linear operator between Banach spaces is bounded iff its graph is closed. {% endtheorem %} We prove a slightly more general version: {% theorem * Closed Graph Theorem (Variant) %} 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. {% endtheorem %} {% proof %} Let us assume first that $T$ is bounded. Let $(x_n,Tx_n)_n$ be a sequence in $\graph{T}$ that converges to some element $(x,y) \in X \times Y$. This means that $x_n \to x$ and $Tx_n \to y$ for $n \to \infty$. The continuity of $T$ implies $Tx_n \to Tx$. Since a convergent series in a Hausdorff space has a unique limit, it follows that $Tx = y$; hence $(x,y)$ lies in $\graph{T}$. This shows that $\graph{T}$ is closed. Conversely, suppose that $\graph{T}$ is a closed subspace of $X \times Y$. Note that $X \times Y$ is a Banach space with norm $\norm{(x,y)} = \norm{x} + \norm{y}$. Therefore $\graph{T}$ is itself as Banach space in the restricted norm $\norm{(x,Tx)} = \norm{x} + \norm{Tx}$. The canonical projections $\pi_X : \graph{T} \to X$ and $\pi_Y : \graph{T} \to Y$ are bounded. Clearly, $\pi_X$ is bijective, so its inverse $\pi_X^{-1} : X \to \graph{T}$ is a bounded operator by the [Bounded Inverse Theorem](/pages/functional-analysis-basics/the-fundamental-four/open-mapping-theorem.html#bounded-inverse-theorem). Consequently the composition, $\pi_Y \circ \pi_X^{-1} : X \to Y$ is bounded. To complete the proof, observe that $\pi_Y \circ \pi_X^{-1} = T$. {% endproof %}