CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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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.

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The dual space separates points of a normed space

Statement

Let $X$ be a normed space over $R$ or $C$ . If $x, y \in X$ with $x \neq y$ , then there exists $f \in X^{*}$ such that $f (x) \neq f (y)$ .

Facts & Assumptions

Given: A normed space $X$ over $R$ or $C$ and two vectors $x, y \in X$ with $x \neq y$ .

[L1]

Every nonzero vector admits a norming functional (Every nonzero vector has a norming functional).

Proof

technique · direct

1.1

Since $x \neq y$ , the vector $x - y$ is nonzero.

given

2.1

By [L1], there exists $f \in X^{*}$ with $f (x - y) = ∥ x - y ∥ .$ In particular $f (x - y) \neq 0$ .

L1step 1.1

3.1

If $f (x) = f (y)$ , then linearity would give $f (x - y) = 0$ , contradicting step 2.1. Therefore $f (x) \neq f (y)$ , so the dual space separates $x$ and $y$ .

step 2.1algebra∎

Depends on

Every nonzero vector has a norming functional

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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

Gerald Teschl, Topics in Real and Functional Analysis, Corollary 4.16 (standard reference, not scraped)
Daniel Daners, Introduction to Functional Analysis, Remark 26.6 (standard reference, not scraped)