LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Basic weak neighborhoods

Statement

For a real or complex normed space $X$ , a weak neighborhood base at $x \in X$ consists of

$U (x; f_{1}, \dots, f_{m}; ε) = {y \in X : ∣ f_{j} (y - x) ∣ < ε (1 \leq j \leq m)}, ε > 0, f_{j} \in X^{*} .$

Here $m = 0$ is allowed and gives $X$ . The weak topology is a locally convex vector topology: addition and joint scalar multiplication are continuous, and the displayed zero neighborhoods are convex and balanced.

Facts & Assumptions

[F1]

Weak topology on a normed space defines the weak topology by inverse images of scalar open sets; finite intersections form a basis.

Proof

Given: $X$ , $x$ , a finite list of bounded scalar-linear functionals, and positive radii.

1.1

Every displayed $U$ is a finite intersection of inverse images of open disks centered at $f_{j} (x)$ , so is weakly open and contains $x$ . Conversely, a finite subbasic intersection containing $x$ contains inverse images of disks of radii $r_{j} > 0$ about $f_{j} (x)$ ; take $ε = \min_{j} r_{j}$ . With no conditions the intersection is $X$ . Thus these sets form a neighborhood base.

F1given

2.1

Put $p (v) = \max_{j} ∣ f_{j} (v) ∣$ , taking $p = 0$ for an empty list. Scalar linearity and the triangle inequality give $p (v + w) \leq p (v) + p (w)$ and $p (a v) = ∣ a ∣ p (v)$ . Consequently ${p < ε}$ is balanced and real-convex. Moreover $p ((v + w) - (v_{0} + w_{0})) < ε$ whenever $p (v - v_{0}), p (w - w_{0}) < ε / 2$ . This proves continuity of addition at every pair.

step 1.1algebra

3.1

At $(a_{0}, v_{0})$ write $a v - a_{0} v_{0} = a (v - v_{0}) + (a - a_{0}) v_{0}$ . Require $∣ a - a_{0} ∣ < \min (1, ε / (2 (p (v_{0}) + 1)))$ and $p (v - v_{0}) < ε / (2 (∣ a_{0} ∣ + 1))$ . Then $p (a v - a_{0} v_{0}) < ε$ . These are product neighborhoods and work also at $a_{0} = 0$ and $v_{0} = 0$ . Thus scalar multiplication is jointly continuous, and the convex zero-neighborhood base proves local convexity. $□$

step 2.1algebra

Depends on

Weak topology on a normed space

Used by

Weak closure of the unit sphere is the closed unit ball Corollary
A weakly convergent net need not be eventually norm bounded Counterexample
Weak closure can exceed sequential weak closure Counterexample
Annihilators are weak and weak star closed Lemma
Continuous dual of a weak topology Theorem
Infinite dimensional weak topology is not first countable Theorem
Weak and norm topologies agree iff finite dimensional Theorem
Weak topology is hausdorff Theorem

Dependency tree · two levels

3 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

Bühler–Salamon, Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)
Teschl, Topics in Real and Functional Analysis (2017); exact harvest in batch coverage (standard reference, not scraped)