PropositionStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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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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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

1.1

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

L1L2

2.1

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.

L2L3algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 64 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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