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Uniformly bounded pointwise-convergent operators converge uniformly on compact sets

Statement

Let $X, Y$ be normed spaces and let $T_{n} : X \to Y$ be bounded linear operators with $M_{0} : = \sup_{n} ∥ T_{n} ∥ < \infty$ . If $T_{n} x \to T x$ for every $x \in X$ , then $T$ is bounded and $T_{n} \to T$ uniformly on every norm-compact subset of $X$ .

Facts & Assumptions

[L1]

For a bounded linear operator, $∥ S x ∥ \leq ∥ S ∥ ∥ x ∥$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum).

[L2]

The maps $T_{n}$ are bounded linear operators (A bounded linear operator between normed spaces).

Proof

technique · direct

Given: The objects and hypotheses in the Statement.

1.1

Passing to limits in the linear identities for [L2] shows that $T$ is [given, L2, L1] linear. By [L1], $∥ T x ∥ = \lim_{n} ∥ T_{n} x ∥ \leq M_{0} ∥ x ∥$ , so $T$ is bounded and $∥ T ∥ \leq M_{0}$ .

L1L2algebra

2.1

Put $M : = M_{0} + ∥ T ∥$ . If $M = 0$ , every $T_{n}$ and $T$ is zero and the result [given, step 1.1] is immediate. Suppose $M > 0$ , fix compact $C$ and $ε > 0$ , and choose a finite $ε / (3 M)$ -net $x_{1}, \dots, x_{r}$ in $C$ .

step 1.1algebra

3.1

Pointwise convergence gives $n_{0}$ such that $∥ (T_{n} - T) x_{j} ∥ < ε / 3$ for all $j$ and $n \geq n_{0}$ . For $x \in C$ choose $j$ with $∥ x - x_{j} ∥ < ε / (3 M)$ . Then [L1] gives

givenL1step 2.1

$∥ (T_{n} - T) x ∥ \leq M ∥ x - x_{j} ∥ + ∥ (T_{n} - T) x_{j} ∥ < 2 ε / 3 < ε .$

Thus convergence is uniform on $C$ . [L1, step 2.1, finite maximum] ∎

Depends on

Used by

A Schauder basis implies the bounded approximation property Theorem

Dependency tree · two levels

6 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

Thomas Schlumprecht, Course Notes in Functional Analysis, Math 655 (standard reference, not scraped)