TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Closed graph theorem

Statement

Assume DC. For Banach spaces $X, Y$ and an everywhere-defined linear $T : X \to Y$ , $T$ is bounded if and only if its graph (The graph of a linear operator with a linear domain) is closed.

Facts & Assumptions

Given: DC, Banach spaces $X, Y$ , and an everywhere-defined linear map $T$ .

Proof

technique · direct

1.1given

If $T$ is bounded, $x_{n} \to x$ and $T x_{n} \to y$ , continuity gives $y = T x$ , so the graph is closed.

1.2given

Conversely, a closed graph is Banach by A closed subspace of a Banach space is Banach, since $X \times Y$ is Banach by Finite products of Banach spaces are Banach.

2.1step 1.2∎

The first projection $G (T) \to X$ is a bounded linear bijection; its inverse is bounded by Bounded inverse theorem. Composing that inverse with the second projection makes $T$ bounded.

Depends on

Used by

An everywhere-defined closed operator on a Banach space is bounded Counterexample
Closed sectorial form and its associated operator Definition
Resolvent and spectrum of a closed operator on a Banach space Definition
Second resolvent identity for a closed perturbation Lemma
Well-posedness of the abstract Cauchy problem is equivalent to generation Theorem

Dependency tree · two levels

13 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Buhler--Salamon, Functional Analysis, Theorem 2.20 (standard reference, not scraped)