ExampleConstruction: Literature-sourcedVerification: AI-generatedverified 2026-09-26 (gpt-6-sol)

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$c_{0}$ is Banach for the supremum norm

Example

Let $c_{0} : = {x = (x_{n})_{n \geq 0} \in ℓ^{\infty} : x_{n} \to 0} .$ With the supremum norm inherited from $ℓ^{\infty}$ , the space $c_{0}$ is Banach.

Facts & Assumptions

Given: An arbitrary $z \in ℓ^{\infty} ∖ c_{0}$ .

[L1]

The space $ℓ^{\infty}$ is Banach for the supremum norm ( $ℓ^{\infty}$ is Banach for the supremum norm).

[L2]

A closed subspace of a Banach space is Banach (A closed subspace of a Banach space is Banach).

Verification

technique · direct

1.1givenalgebra

Since $z_{n}$ does not tend to zero, there is $ε > 0$ such that for every $N$ some $n \geq N$ satisfies $∣ z_{n} ∣ \geq ε$ .

2.1step 1.1choosealgebra

For any $y \in c_{0}$ , choose $N$ so that $∣ y_{n} ∣ < ε / 2$ for all $n \geq N$ , and then choose $n \geq N$ as in step 1.1. Thus $∥ z - y ∥_{\infty} \geq ∣ z_{n} ∣ - ∣ y_{n} ∣ \geq ε / 2$ . The open ball of radius $ε / 2$ about $z$ misses $c_{0}$ , so $c_{0}$ is closed.

3.1step 2.1L1L2algebra∎

The limit laws for scalar sequences make $c_{0}$ a linear subspace of $ℓ^{\infty}$ . Since it is closed by step 2.1 and $ℓ^{\infty}$ is Banach by [L1], [L2] makes $c_{0}$ Banach.

Depends on

Used by

Dependency tree · two levels

5 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)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis (standard reference, not scraped)

c0c0​ is Banach for the supremum norm