A linear operator between Banach spaces is bounded if and only if its graph is a closed subset of the product space. It offers a practical route to proving continuity of a linear map without checking boundedness directly.
Facts
StatementA linear map between two complete normed spaces, such as Banach spaces, is continuous if and only if its graph is closed in the product space. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Sources
1. Closed Graph Theorem (Wikipedia)
Wikimedia FoundationLead section, theorem statement sentenceQuote, Lead section, theorem statement sentence
A linear map between two F-spaces (e.g. Banach spaces) is continuous if and only if its graph is closed.
Closed Graph Theorem (MathWorld)
Theorem statementQuote, Theorem statement
The closed graph theorem states that a linear operator between two Banach spaces X and Y is continuous iff it has a closed graph, where the "graph" {(x,f(x)):x in X} is considered closed if it is a closed subset of X×Y equipped with the product topology.
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