PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-16

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.

On $R^{n}$ , abstract Cauchy–Schwarz is exactly the published finite-sum Cauchy–Schwarz inequality

Statement

For $a, b \in R^{n}$ , Cauchy–Schwarz in the standard coordinate inner product is exactly

$∣ \sum_{k < n} a_{k} b_{k} ∣ \leq \sqrt{\sum_{k < n} a_{k}^{2}} \sqrt{\sum_{k < n} b_{k}^{2}},$

with equality exactly when the two lists are proportional in the symmetric sense. This includes $n = 0$ .

Facts & Assumptions

Given: Real coordinate vectors $a, b \in R^{n}$ .

[L1]

The standard real coordinate pairing is $⟨ a, b ⟩ = \sum_{k < n} a_{k} b_{k}$ , with $∥ a ∥^{2} = \sum_{k < n} a_{k}^{2}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

[L2]

Abstract Cauchy–Schwarz has equality exactly for linearly dependent vectors (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

[L3]

The published finite-sum theorem states the displayed inequality and equality exactly when some $(λ, μ) \neq (0, 0)$ satisfies $λ a_{k} = μ b_{k}$ for every $k < n$ (The Cauchy-Schwarz inequality for finite sums).

Proof

technique · direct

1.1L1L2

Substituting [L1] into [L2] gives the displayed finite-sum inequality term for term.

2.1L2L3algebra∎

Coordinate vectors are linearly dependent exactly when there is a nonzero scalar pair $(λ, μ)$ with $λ a_{k} = μ b_{k}$ for all $k$ , so the equality condition agrees with [L3]. For $n = 0$ , both sides are zero and the empty lists satisfy the symmetric proportionality condition.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

23 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

Sheldon Axler, Linear Algebra Done Right, 4th ed., §6A (standard reference, not scraped)

On RnRn, abstract Cauchy–Schwarz is exactly the published finite-sum Cauchy–Schwarz inequality