PropositionStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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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.

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Complexification is a functor on real vector spaces and real-linear maps

Statement

For real vector spaces $V, W, U$ and real-linear maps $T : V \to W$ and $S : W \to U$ , the complexification of Complexification of a real-linear map satisfies

$({id}_{V})_{C} = {id}_{V_{C}}, (S \circ T)_{C} = S_{C} \circ T_{C} .$

Thus $V \mapsto V_{C}$ , $T \mapsto T_{C}$ is a functor from real vector spaces to complex vector spaces.

Facts & Assumptions

Given: Real-linear maps $T : V \to W$ and $S : W \to U$ .

[L1]

The complexification of a real-linear map is $T_{C} = {id}_{C} \otimes T$ , with $T_{C} (z \otimes v) = z \otimes T (v)$ (Complexification of a real-linear map).

[L2]

For homomorphisms of right and left modules, $id \otimes id = id$ and $(f^{'} \circ f) \otimes (g^{'} \circ g) = (f^{'} \otimes g^{'}) \circ (f \otimes g)$ (Module homomorphisms induce tensor-product homomorphisms functorially).

Proof

technique · direct

1.1

By [L2] applied with $f = f^{'} = {id}_{C}$ and $g = {id}_{V}$ , one has $({id}_{V})_{C} = {id}_{C} \otimes {id}_{V} = {id}_{V_{C}}$ by [L1].

L1L2

1.2

By [L2] applied with $f = f^{'} = {id}_{C}$ , $g = T$ and $g^{'} = S$ , one has $(S \circ T)_{C} = {id}_{C} \otimes (S \circ T) = ({id}_{C} \circ {id}_{C}) \otimes (S \circ T) = ({id}_{C} \otimes S) \circ ({id}_{C} \otimes T) = S_{C} \circ T_{C}$ by [L1].

L1L2algebra

2.1

Steps 1.1 and 1.2 are exactly the identity and composition laws of a functor.

step 1.1step 1.2∎

Depends on

Used by

Dependency tree · two levels

9 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

Keith Conrad, Complexification (notes) (standard reference, not scraped)