False statementConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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FALSE: complexification doubles finite dimension

Statement

Complexification doubles finite dimension: for every finite-dimensional real vector space $V$ ,

$\dim_{C} (C \otimes_{R} V) = 2 \dim_{R} V .$

Facts & Assumptions

Given: A finite-dimensional real vector space $V$ and its complexification $V_{C}$ .

[L1]

The complexification of $R^{n}$ is canonically $C^{n}$ through the standard inclusion (The standard embedding $R^{n} ↪ C^{n}$ is the canonical complexification map).

[L2]

A real basis becomes a complex basis after complexification, so $\dim_{C} (C \otimes_{R} V) = \dim_{R} V$ (A real basis becomes a complex basis after complexification, so $\dim_{C} (C \otimes_{R} V) = \dim_{R} V$ ).

Refutation

technique · direct

1.1L2

By [L2], $\dim_{C} V_{C} = \dim_{R} V$ for every finite-dimensional real $V$ : the embedded image of a real basis is already a complex basis of the complexification.

1.2L1L2

The concrete witness $V = R$ confirms the correct value: by [L1] with $n = 1$ , $V_{C} ≅ C$ , so $\dim_{C} V_{C} = 1 = \dim_{R} R$ , not $2$ .

2.1step 1.1step 1.2

The doubling behaviour belongs to realification, the reverse construction, which replaces complex scalars by real ones; complexification keeps the numerical dimension unchanged.

3.1step 1.1step 1.2∎

Steps 1.1 and 1.2 contradict the claimed factor of $2$ , so the displayed statement is false.

Depends on

Used by

Nothing in the library uses this result yet.

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

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