TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-07

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Baire dichotomy for a pointwise-defined family of bounded linear operators

Statement

Assume DC. Let $X$ be Banach, $Y$ normed, and $F \subseteq B (X, Y)$ (A bounded linear operator between normed spaces). Either $\sup_{T \in F} ∥ T ∥ < \infty$ (The operator norm as the least bound and as the unit-sphere or unit-ball supremum), or

$S : = {x \in X : \sup_{T \in F} ∥ T x ∥ = \infty}$

is a dense $G_{δ}$ subset of $X$ .

Facts & Assumptions

Given: DC and $X, Y, F$ as in the statement.

Proof

technique · direct

1.1

Let $E_{n} = {x : \sup_{T} ∥ T x ∥ \leq n}$ . These sets are closed, and $X ∖ S = ⋃_{n} E_{n}$ .

given

2.1

If some $E_{n}$ has nonempty interior, the translation-and-rescaling argument of Uniform boundedness principle gives a common operator-norm bound.

step 1.1

2.2

Otherwise every $E_{n}$ is closed with empty interior. Its complement is open dense, so $S = ⋂_{n} (X ∖ E_{n})$ is $G_{δ}$ and dense by Under Dependent Choice, a nonempty complete metric space is not a countable union of closed sets with empty interior.

step 1.1

3.1

The two alternatives exhaust the cases from step 2.1, proving the dichotomy.

step 2.1step 2.2∎

Depends on

Used by

Condensation of singularities Example

Dependency tree · two levels

10 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

Teschl, Topics in Real and Functional Analysis, Theorem 4.3 (standard reference, not scraped)