CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-06

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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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Closed finite-codimensional subspaces are complemented

Statement

Every closed finite-codimensional linear subspace $M$ of a normed space $X$ is complemented in $X$ .

Facts & Assumptions

Given: A closed subspace $M \subseteq X$ with finite-dimensional $X / M$ .

[F1]

The quotient seminorm is a norm exactly when the subspace is closed (The quotient seminorm is a norm exactly when the subspace is closed).

[F2]

Coordinate functionals for a fixed basis of a finite-dimensional normed space are bounded (A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space).

[F3]

The range of a bounded projection is complemented (A closed subspace is complemented exactly when it is the range of a bounded projection).

Proof

technique · direct

1.1

By [F1], $Q = X / M$ is a normed finite-dimensional quotient. Choose a basis $q_{1}, \dots, q_{n}$ of $Q$ , representatives $x_{j} \in X$ , and bounded coordinate maps $ϕ_{j}$ supplied by [F2].

F1F2givenchoose

2.1

The map $S : Q \to X$ , $S (q) = \sum_{j} ϕ_{j} (q) x_{j}$ , is bounded and satisfies $π S = I_{Q}$ . Hence $P = I_{X} - S π$ is bounded, $P^{2} = P$ , and $ran P = M$ .

step 1.1algebra

3.1

By [F3], $M$ is complemented.

step 2.1F3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

Piotr Hajlasz, Functional Analysis, Theorem 10.17(b) (standard reference, not scraped)