LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generatedaudited 2026-09-06

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c_0 is a closed subspace of ell-infinity

Statement

$c_{0}$ is a closed linear subspace of $ℓ^{\infty}$ in the sup norm.

Facts & Assumptions

Given: The sequence spaces $c_{0} \subseteq ℓ^{\infty}$ .

[F1]

$c_{0}$ consists of bounded sequences tending to zero, with the sup norm (The sequence spaces c_0 and ell-infinity).

Proof

technique · direct

1.1F1given

Sums and scalar multiples of null sequences are null, so $c_{0}$ is a linear subspace.

1.2F1givenchoose

If $x^{(j)} \in c_{0}$ and $∥ x^{(j)} - x ∥_{\infty} \to 0$ , choose $j$ with this norm below $ε / 2$ , then $N$ with $∣ x_{n}^{(j)} ∣ < ε / 2$ for $n \geq N$ . Thus $∣ x_{n} ∣ < ε$ for $n \geq N$ .

2.1step 1.2∎

Hence $x \in c_{0}$ and the subspace is closed.

Depends on

Used by

A closed subspace of ell-infinity that is not complemented Counterexample
The transpose of an injective map need not have norm-dense range Counterexample
A closed uncomplemented subspace Example
Reflexivity of ℓᵖ and Lᵖ Example
The quotient ell-infinity/c₀ has no countable separating family Lemma
c₀ is not complemented in ell-infinity Theorem

Dependency tree · two levels

7 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

Piotr Hajlasz, Functional Analysis, §10.5 (standard reference, not scraped)