DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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 $\mathcal L(V,W)$ of linear maps with pointwise addition and scalar multiplication

Definition

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

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

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

$(S+T)(v):=S(v)+T(v),\qquad (\lambda T)(v):=\lambda T(v).$

The zero function $v\mapsto0_W$ is denoted by $0$ . Closure and the vector-space axioms for these operations are proved in $\mathcal L(V,W)$ is a vector space over the common scalar field ↗.

Depends on

Used by

mathcal L(V,W) is a vector space over the common scalar field Proposition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 12 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

S. Schiavone, MIT 18.700 Day 9, §II.3.1 (standard reference, not scraped)

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