The Closed Graph Theorem is a powerful tool in functional analysis. It links the continuity of linear operators between Banach spaces to the closure of their graphs, simplifying proofs of operator continuity in many cases.

This theorem has wide-ranging applications in functional analysis. It's used to prove the continuity of inverse operators, establish the Banach Isomorphism Theorem, and show continuity in various mathematical settings like Fourier transforms.

The Closed Graph Theorem

Closed Graph Theorem formulation

• States if $X$ and $Y$ are Banach spaces (complete normed vector spaces) and $T: X \to Y$ is a linear operator with a closed graph $G(T) = {(x, Tx) : x \in X}$ in the product topology of $X \times Y$, then $T$ is continuous
• Requires $X$ and $Y$ to be Banach spaces (e.g., $L^p$ spaces, $C[a,b]$) and $T$ to be a linear operator between them (e.g., differentiation, integration)
• Graph of $T$, $G(T)$, must be closed in $X \times Y$ equipped with the product topology (e.g., pointwise convergence, uniform convergence)

Proof using Open Mapping Theorem

• Relies on the Open Mapping Theorem which states a surjective bounded linear operator $T: X \to Y$ between Banach spaces is an open map
• Defines a new linear, bounded, and surjective operator $S: G(T) \to X$ by $S(x, Tx) = x$
• Applies the Open Mapping Theorem to show $S$ is an open map
• Uses the openness of $S$ to prove $T$ is continuous
  1. For any open set $U \subset X$, $S^{-1}(U) = {(x, Tx) : x \in U}$ is open in $G(T)$
  2. Since $G(T)$ is closed in $X \times Y$, $S^{-1}(U)$ is open in $X \times Y$
  3. The projection of $S^{-1}(U)$ onto $Y$ is $T(U)$, which is open in $Y$
  4. Thus, $T$ maps open sets to open sets, implying continuity

Continuity of linear operators

• Proves a linear operator $T: X \to Y$ is continuous by showing its graph $G(T)$ is closed
• Commonly used to prove continuity of the inverse operator $T^{-1}$ when $T$ is bijective by showing $G(T^{-1}) = {(Tx, x) : x \in X}$ is closed
• Establishes continuity of linear operators defined by specific properties (e.g., proving a linear functional $f: X \to \mathbb{R}$ satisfying a certain condition is continuous)

Implications in Banach spaces

• Provides a topological characterization of continuous linear operators between Banach spaces a linear operator is continuous if and only if its graph is closed
• Helps prove the Banach Isomorphism Theorem if $T: X \to Y$ is a bijective continuous linear operator between Banach spaces, then $T^{-1}$ is also continuous
• Establishes continuity of linear operators in various settings (e.g., proving continuity of the Fourier transform on certain function spaces)
• Closely related to other fundamental results in functional analysis such as the Open Mapping Theorem and the Uniform Boundedness Principle