DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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Banach space

Definition

Let $V$ be a normed space in the sense of A norm on a real vector space, the induced metric, and the dictionary with the metric axioms, with induced metric $d (x, y) = ∥ x - y ∥$ . Then $V$ is a Banach space when this metric space is complete in the sense of Complete metric space: every Cauchy sequence converges in the space.

Equivalently: every Cauchy sequence in $V$ for the norm metric converges to a point of $V$ .

Remarks

The metric is part of the data. Completeness is always completeness for the metric induced by the named norm.
By Real and complex scalar conventions for normed spaces, the same definition is used over $R$ and over $C$ .

Depends on

Used by

Completion of a normed space Definition
C_b(X) is Banach for the supremum norm Example
ℓ^∞ is Banach for the supremum norm Example
A closed subspace of a Banach space is Banach Lemma
A complete normed subspace is closed Lemma
The classical Lᵖ spaces are Banach spaces Remark
Bounded linear maps extend uniquely across the completion Theorem
Finite products of Banach spaces are Banach Theorem
Series criterion for Banach spaces Theorem
The metric completion of a normed space carries a unique compatible Banach-space structure Theorem

Dependency tree · two levels

19 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

Theo Buhler and Dietmar A. Salamon, Functional Analysis (standard reference, not scraped)
Gerald Teschl, Topics in Real and Functional Analysis (standard reference, not scraped)