LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A closed subspace of a Banach space is Banach

Statement

Let $V$ be a Banach space and let $W \subseteq V$ be a closed linear subspace, equipped with the restricted norm. Then $W$ is a Banach space.

Facts & Assumptions

Given: A Banach space $V$ and a normed subspace $W \subseteq V$ .

[L1]

The metric of a normed subspace is the ambient metric restricted to the subspace (Normed subspace).

[L2]

In a complete metric space, every closed subspace is complete (A subspace of a complete metric space is complete iff it is closed, and a complete subspace of any metric space is closed).

[L3]

A Banach space is a normed space complete for its norm metric (Banach space).

Proof

technique · direct

1.1

By [L3], the ambient norm metric on $V$ is complete.

1.2

By [L1], the restricted norm on $W$ induces exactly the ambient subspace metric on $W$ .

2.1

Since $W$ is closed in the complete metric space $V$ , [L2] makes that subspace metric complete.

step 1.1step 1.2L2

3.1

Completeness of the restricted norm metric is exactly the Banach property for $W$ by [L3].

step 2.1L3∎

Depends on

Used by

c₀ is Banach for the supremum norm Example

Dependency tree · two levels

16 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)