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

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.

Finite-dimensional subspaces are complemented

Statement

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

Facts & Assumptions

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

[F1]

Relative to a fixed finite basis, every coordinate functional on a finite-dimensional normed space is bounded (A chosen algebraic basis identifies a finite-dimensional normed space with a coordinate space).

[F2]

A bounded functional on a subspace extends norm-preservingly to the ambient normed space (A bounded linear functional on an arbitrary subspace extends with the same norm, without assuming the subspace is closed).

[F3]

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

Proof

technique · direct

1.1

Choose a basis $e_{1}, \dots, e_{n}$ of $M$ , and let $ϕ_{j} : M \to K$ be its coordinate maps. By [F1]--[F2], extend each $ϕ_{j}$ to $f_{j} \in X^{*}$ .

givenF1F2choose

2.1

Define $P : X \to X$ by $P (x) = \sum_{j = 1}^{n} f_{j} (x) e_{j}$ . It is bounded, has range in $M$ , and for $m = \sum a_{j} e_{j} \in M$ satisfies $P (m) = m$ .

step 1.1algebra

3.1

Thus $P^{2} = P$ and $ran P = M$ ; [F3] makes $M$ complemented.

step 2.1F3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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(a) (standard reference, not scraped)