Open Mapping and Closed Graph Theorems - Key Properties

The Closed Graph Theorem provides a powerful criterion for boundedness of linear operators, transforming a sequential condition into a norm estimate.

TheoremClosed Graph Theorem

Let $X$ and $Y$ be Banach spaces and $T : X \to Y$ a linear operator with closed graph. Then $T$ is bounded.

This theorem is extremely useful in practice because verifying that a graph is closed (a sequential condition) is often much easier than directly estimating the operator norm. The completeness of both $X$ and $Y$ is essential.

Proof

The graph $\Gamma(T) = \{(x, T(x)) : x \in X\}$ is a closed subspace of $X \times Y$ . With the product norm $\|(x,y)\| = \|x\|_X + \|y\|_Y$ , the space $X \times Y$ is a Banach space, so $\Gamma(T)$ is also a Banach space.

The projection $\pi_1 : \Gamma(T) \to X$ defined by $\pi_1(x, T(x)) = x$ is a bijective bounded linear operator. By the Open Mapping Theorem, $\pi_1^{-1}$ is bounded.

For any $x \in X$ , we have $\pi_1^{-1}(x) = (x, T(x))$ , so $\|(x, T(x))\| = \|\pi_1^{-1}(x)\| \leq \|\pi_1^{-1}\| \|x\|$ $\|x\| + \|T(x)\| \leq \|\pi_1^{-1}\| \|x\|$ $\|T(x)\| \leq (\|\pi_1^{-1}\| - 1) \|x\|$

Therefore $T$ is bounded.

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ExampleApplications

Differential Operators: The operator $D : H^1(\Omega) \to L^2(\Omega)$ given by $D(u) = \nabla u$ has closed graph, hence is bounded
Multiplication Operators: If $\phi \in L^\infty$ , then $M_\phi : L^2 \to L^2$ defined by $(M_\phi f)(x) = \phi(x) f(x)$ has closed graph
Weak Derivatives: If $u_n \to u$ in $L^2$ and $Du_n \to v$ in $L^2$ , then $Du = v$ (in the weak sense)
Automatic Continuity: Many naturally defined operators are automatically bounded by closed graph arguments

TheoremEquivalent Formulations

For a linear operator $T : X \to Y$ between Banach spaces, the following are equivalent:

$T$ is bounded
$T$ has a closed graph
If $x_n \to 0$ in $X$ and $T(x_n) \to y$ in $Y$ , then $y = 0$

Remark

The Closed Graph Theorem fails if either $X$ or $Y$ is not complete. For instance, on incomplete spaces, there exist unbounded operators with closed graphs. Completeness is absolutely essential.

This theorem is indispensable in PDE theory, where verifying continuity of solution operators directly can be extremely difficult, but showing closed graph is often straightforward using integration by parts and weak convergence.