Alphabeta Math
PropositionStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 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.

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

Statement

If V and W are vector spaces over a field F, then L(V,W) is a vector space over F under pointwise addition and scalar multiplication.

Facts & Assumptions

Given: Vector spaces V,W over the same field F and the pointwise operations on L(V,W).

[L1]

L(V,W) is the set of linear maps, with pointwise addition and scalar multiplication (The space L(V,W) of linear maps with pointwise addition and scalar multiplication).

Proof

technique · direct
1.1givenL1

If S,T are linear, then (S+T)(au+bv)=a(S+T)(u)+b(S+T)(v), and if λ∈F, then (λT)(au+bv)=a(λT)(u)+b(λT)(v); hence the pointwise operations remain inside L(V,W).

2.1step 1.1L1

Evaluating at an arbitrary v∈V reduces associativity, commutativity, both distributive laws, scalar associativity, and the scalar identity law to the corresponding vector-space laws in W.

3.1step 2.1L1∎

The zero function is the additive identity and (−T)(v):=−T(v) is the additive inverse of T. The same formulas cover the zero domain and zero codomain, so all vector-space axioms hold.

Depends on

Dependency tree · 2 levels

Sources