DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-01

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.

A complemented closed subspace of a normed space

Definition

Let $X$ be a normed space and let $M \leq X$ be a closed linear subspace. We say that $M$ is complemented when there is a closed linear subspace $N \leq X$ such that every $x \in X$ admits a unique decomposition

$x = m + n (m \in M, n \in N),$

and the coordinate maps

$P_{M} (x) : = m, P_{N} (x) : = n$

are bounded linear operators on $X$ .

Remarks

The decomposition is written $X = M \oplus N$ .
The theorem below shows that this is equivalent to $M$ being the range of a bounded projection.

A complemented closed subspace of a normed space

Definition

Remarks

Depends on

Dependency tree · 2 levels

Sources