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$C_{b} (X)$ is Banach for the supremum norm

Example

Let $X$ be a nonempty topological space, and let $C_{b} (X)$ be the space of bounded continuous scalar-valued functions on $X$ with norm $∥ f ∥_{\infty} : = \sup_{x \in X} ∣ f (x) ∣ .$ Then $C_{b} (X)$ is a Banach space.

Facts & Assumptions

Given: A nonempty topological space $X$ and a Cauchy sequence $(f_{m})$ in $C_{b} (X)$ for the supremum norm.

[L1]

Uniform limits of continuous functions are continuous (A uniform limit of continuous functions is continuous, so $C (X, Y)$ is closed in $Y^{X}$ under the uniform metric).

[L2]

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

Verification

technique · direct

1.1givenconstruct

For each $x \in X$ , the scalar sequence $(f_{m} (x))_{m}$ is Cauchy because $∣ f_{m} (x) - f_{ℓ} (x) ∣ \leq ∥ f_{m} - f_{ℓ} ∥_{\infty}$ . Define $f (x) : = \lim_{m} f_{m} (x)$ .

2.1step 1.1given

Choose $M$ with $∥ f_{m} - f_{ℓ} ∥_{\infty} < 1$ for $m, ℓ \geq M$ . Fixing $ℓ = M$ and letting $m \to \infty$ pointwise gives $∣ f (x) ∣ \leq ∥ f_{M} ∥_{\infty} + 1$ for every $x$ , so $f$ is bounded.

2.2step 1.1givenL1

Given $ε > 0$ , choose $M$ with $∥ f_{m} - f_{ℓ} ∥_{\infty} < ε$ for $m, ℓ \geq M$ . Letting $ℓ \to \infty$ pointwise yields $∥ f_{m} - f ∥_{\infty} \leq ε$ for $m \geq M$ . Thus $f_{m} \to f$ uniformly, and [L1] makes $f$ continuous.

3.1step 2.1step 2.2L2∎

So every supremum-norm Cauchy sequence in $C_{b} (X)$ converges in $C_{b} (X)$ , and [L2] shows that $C_{b} (X)$ is Banach.

Depends on

Used by

C(K) is Banach when K is compact metric Example

Dependency tree · two levels

26 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)

Cb(X)Cb​(X) is Banach for the supremum norm