DefinitionDefinition: AI-adaptedProof: AI-generatedPipeline-generatedprecheck pass

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 complex normed linear subspace

Definition

Let $(V, N)$ be a complex normed space (Complex norm and complex normed space) and let $W \subseteq V$ be a complex-linear subspace (Linear subspace of a vector space over $C$ ). Define the restricted norm $N_{W} : W \to R$ by $N_{W} (w) : = N (w)$ . The pair $(W, N_{W})$ is the complex normed subspace determined by $W$ .

Facts & Assumptions

Given: A complex normed space $(V, N)$ and a complex-linear subspace $W \subseteq V$ .

[F1]

$N$ separates zero, satisfies $N (λ v) = ∣ λ ∣_{C} N (v)$ for every $λ \in C$ , and satisfies the triangle inequality (Complex norm and complex normed space).

[F2]

A linear subspace contains zero, is closed under addition, and is closed under multiplication by every scalar in its field; here that field is $C$ (Linear subspace of a vector space).

Proof

1.1F1

For $w \in W$ , $N_{W} (w) = 0$ iff $N (w) = 0$ iff $w = 0$ by [F1], so the restriction separates zero.

2.1F1F2step 1.1∎

If $w \in W$ and $λ \in C$ , then $λ w \in W$ by [F2] and $N_{W} (λ w) = N (λ w) = ∣ λ ∣_{C} N (w) = ∣ λ ∣_{C} N_{W} (w)$ by [F1]. If $u, w \in W$ , then $u + w \in W$ and $N_{W} (u + w) = N (u + w) \leq N (u) + N (w) = N_{W} (u) + N_{W} (w)$ , again by [F1]. Thus the restricted function has the full complex homogeneity and triangle properties as well.

Depends on

Used by

Dependency tree · two levels

8 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

Tomasz Kochanek, Functional analysis, Lecture 1 (complete five-page lecture notes) (standard reference, not scraped)
M. Winklmeier, Functional Analysis lecture notes (complete 140-page lecture notes) (standard reference, not scraped)