TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedverified 2026-09-24 (gpt-6-sol)

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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.

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

Closed hyperplanes are kernels of nonzero functionals

Statement

A linear hyperplane $H \subseteq X$ is closed if and only if $H = \ker f$ for some nonzero $f \in X^{*}$ .

Facts & Assumptions

Given: A linear hyperplane $H \subseteq X$ .

[F1]

Since a linear hyperplane has codimension one, for any $x_{0} \notin H$ each $x \in X$ has a unique decomposition $x = h + t x_{0}$ with $h \in H$ and $t \in K$ (Linear hyperplane).

[F2]

Since $H$ is closed, $x_{0} \notin H$ implies $δ : = dist (x_{0}, H) > 0$ (The closure of a nonempty $A$ is ${x : d (x, A) = 0}$ , equals $A$ together with its limit points, and is the smallest closed superset).

Proof

technique · direct

1.1givenF1chooseconstruct

Suppose $H$ is closed and choose $x_{0} \notin H$ . By [F1], define $f (h + t x_{0}) : = t$ on $X$ . Uniqueness of the decomposition makes $f$ well defined and linear, with $f (x_{0}) = 1$ and $\ker f = H$ .

2.1F1F2step 1.1algebra

By [F2], $δ = dist (x_{0}, H) > 0$ . For $x = h + t x_{0}$ with $t \neq 0$ , the vector $x / t = x_{0} + h / t$ has norm at least $δ$ , since $- h / t \in H$ . Hence $∣ f (x) ∣ = ∣ t ∣ \leq δ^{- 1} ∥ x ∥$ ; the same bound is clear for $t = 0$ . Thus $f$ is bounded and nonzero, so $f \in X^{*}$ and $H = \ker f$ .

3.1givenalgebra∎

Conversely, if $H = \ker f$ with $f \in X^{*}$ nonzero, continuity makes $H$ closed. The induced nonzero map $X / H \to K$ is injective and onto, so $\dim (X / H) = 1$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

22 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 Buehler and Dietmar Salamon, Functional Analysis, Exercise 2.47 (standard reference, not scraped)