DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

The space $L (V, W)$ of linear maps with pointwise addition and scalar multiplication

Let $V$ and $W$ be vector spaces over the same field $F$ . Write

$L (V, W) : = {T : V \to W : T is linear} .$

For $S, T \in L (V, W)$ and $λ \in F$ , define pointwise operations by

$(S + T) (v) : = S (v) + T (v), (λ T) (v) : = λ T (v) .$

The zero function $v \mapsto 0_{W}$ is denoted by $0$ . Closure and the vector-space axioms for these operations are proved in $L (V, W)$ is a vector space over the common scalar field ↗.

Depends on