The Closed Graph Theorem — Mathematics Knowledge System

Overview

The closed graph theorem states that a linear operator between Banach spaces is bounded (continuous) if and only if its graph is a closed subset of the product space. This provides a practical criterion for boundedness: instead of finding a norm estimate ||Tx|| <= C||x||, one only needs to verify that x_n -> x and Tx_n -> y implies Tx = y. The theorem is a direct consequence of the open mapping theorem.

Intuition

A map has a closed graph if 'the limit of outputs is the output of the limit': whenever inputs x_n converge to x and outputs Tx_n converge to some y, we must have y = Tx. In finite dimensions, every linear map is continuous, so every linear map has a closed graph. In infinite dimensions, closed graph is weaker than boundedness for arbitrary maps — but the theorem says that for linear maps between Banach spaces, closed graph is equivalent to boundedness.

Formal Definition

Definition

The graph of T: X -> Y is the set G(T) = {(x, Tx) : x in X} in X x Y. We say T is a closed operator if G(T) is a closed subspace of X x Y (with the product topology). For a linear operator between Banach spaces, closed graph is equivalent to bounded (continuous).

G(T) = \{(x, Tx) : x \in X\} \subseteq X \times Y

Graph of T

G(T) \text{ closed} \iff (x_n \to x,\; Tx_n \to y) \implies y = Tx

Closed graph condition

\|( x, y)\|_{X \times Y} = \|x\|_X + \|y\|_Y

Product norm

Notation

Notation	Meaning
$G(T)$	Graph of the operator T
$X \times Y$	Product space with norm \|\|(x,y)\|\| = \|\|x\|\| + \|\|y\|\|

Theorems

Theorem 7.1: Closed Graph Theorem

Let X and Y be Banach spaces and T: X -> Y a linear operator. T is bounded if and only if its graph G(T) is closed in X x Y.

Theorem 7.2: Closed Graph Theorem (proof via Open Mapping)

If T: X -> Y is linear with closed graph, define pi_1: G(T) -> X by pi_1(x,Tx) = x. Then pi_1 is bijective and bounded, hence has bounded inverse by the open mapping theorem. Since pi_2(x,Tx) = Tx = pi_2 composed with pi_1^{ -1 }, T is bounded.

Theorem 7.3: Hellinger-Toeplitz Theorem

A symmetric operator T: H -> H on a Hilbert space H satisfying <Tx,y> = <x,Ty> for all x,y in H is necessarily bounded.

Worked Examples

1
Suppose f_n -> f in (C^1[0,1], ||·||_{C^1}) and Df_n = f_n' -> g in (C[0,1], ||·||_inf).
2
Since ||f_n - f||_{C^1} -> 0, we have ||f_n' - f'||_inf -> 0, so f_n' -> f' uniformly.
3
By uniqueness of uniform limits, g = f'. Thus D(f) = g.
4
The graph of D is closed. By the closed graph theorem, D is bounded.

✓ Answer

D is bounded from (C^1[0,1], ||·||_{C^1}) to C[0,1] by the closed graph theorem.

Practice Problems

Mediumproof writing

Prove the Hellinger-Toeplitz theorem: if T: H -> H is everywhere-defined on a Hilbert space and <Tx,y> = <x,Ty> for all x,y, then T is bounded.

MediumMultiple choice

Which space must be complete for the closed graph theorem to apply?

Only the domain XOnly the codomain YBoth X and Y must be Banach spacesNeither; the theorem holds for all normed spaces

Mediumfree response

Explain the relationship between the closed graph theorem and the open mapping theorem. Can one be derived from the other?

Common Mistakes

Common Mistake

Thinking a closed graph always implies boundedness regardless of the spaces involved.

The closed graph theorem requires both spaces to be Banach. For incomplete spaces, there are linear operators with closed graphs that are not bounded.

Common Mistake

Confusing 'closed operator' with 'closed-range operator'.

A closed operator has a closed graph (the condition involves convergence of both inputs and outputs). A closed-range operator has T(X) closed in Y. These are different properties.

Quiz

Historical Background

The closed graph theorem was proved by Stefan Banach in his 1932 monograph Théorie des opérations linéaires, as a corollary of the open mapping theorem. It quickly became a standard tool for establishing the boundedness of differential operators and other naturally defined linear maps in analysis. John von Neumann used it heavily in his formulation of quantum mechanics.

1932
Banach proves the closed graph theorem in his monograph
Stefan Banach
1932
Von Neumann applies closed operators in quantum mechanics
John von Neumann
1950s
Closed graph theorem generalised to locally convex spaces by Pták and others
Vlastimil Pták

Summary

The closed graph theorem: a linear operator between Banach spaces is bounded iff its graph is closed.
Closed graph means: x_n -> x and Tx_n -> y implies y = Tx (limit of outputs = output of limit).
The theorem is a corollary of the open mapping theorem, applied to the projection from G(T) to X.
Hellinger-Toeplitz: a symmetric everywhere-defined operator on a Hilbert space is bounded.
Unbounded operators (like differential operators) can only be closed if their domain is restricted (they are not everywhere defined).

References

BookRudin, W. — Functional Analysis (2nd ed.), Chapter 2
BookConway, J. — A Course in Functional Analysis, Chapter 3
WebsiteWikipedia: Closed graph theorem