ExampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passaudited 2026-08-29

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.

Realifying $C^{n}$ gives $R^{2 n}$ with basis $e_{1}, i e_{1}, \dots, e_{n}, i e_{n}$

Example

The realification $(C^{n})_{R}$ of the complex coordinate space has real dimension $2 n$ . Writing the standard complex basis vectors as $e_{1}, \dots, e_{n}$ , an explicit real basis is

$e_{1}, i e_{1}, e_{2}, i e_{2}, \dots, e_{n}, i e_{n},$

so $(C^{n})_{R} ≅ R^{2 n}$ by sending $e_{j}$ to $e_{2 j - 1}$ and $i e_{j}$ to $e_{2 j}$ of $R^{2 n}$ .

Facts & Assumptions

Given: The complex vector space $C^{n}$ with standard basis $e_{1}, \dots, e_{n}$ .

[L1]

The realification $W_{R}$ is the real vector space with the same underlying set and addition as $W$ , and scalar multiplication restricted to $R \subseteq C$ (Realification of a complex vector space by restriction of scalars).

[L2]

If $W$ is finite-dimensional over $C$ with $\dim_{C} W = n$ , then $\dim_{R} W_{R} = 2 n$ (Realification doubles finite dimension).

Verification

technique · direct

1.1

The standard basis has $n$ elements, so $\dim_{C} C^{n} = n$ .

given

1.2

The displayed list spans $(C^{n})_{R}$ : every $w = \sum_{j} z_{j} e_{j}$ writes $z_{j} = a_{j} + i b_{j}$ with real $a_{j}, b_{j}$ , and by [L1] scalar multiplication by the real parts is real scalar multiplication, giving $w = \sum_{j} a_{j} e_{j} + \sum_{j} b_{j} (i e_{j})$ .

L1algebra

2.1

By [L2], $\dim_{R} (C^{n})_{R} = 2 n$ .

L2step 1.1

2.2

The list is real-linearly independent: $\sum_{j} a_{j} e_{j} + \sum_{j} b_{j} (i e_{j}) = 0$ means $\sum_{j} (a_{j} + i b_{j}) e_{j} = 0$ in $C^{n}$ , so complex independence of $(e_{1}, \dots, e_{n})$ forces every $a_{j} = b_{j} = 0$ .

step 1.2algebra

3.1

Steps 1.2 and 2.2 exhibit the displayed list as a real basis with $2 n$ entries, matching the dimension of step 2.1; the coordinate map identifies it with $R^{2 n}$ .

step 1.2step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Mikhail Troshkin, Real-complex linear algebra and abelian varieties (standard reference, not scraped)

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