CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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.

Cauchy-Schwarz for random variables

Statement

If $X, Y \in L^{2} (P)$ are real random variables, then $E [∣ X Y ∣] \leq (E [X^{2}])^{1 / 2} (E [Y^{2}])^{1 / 2} .$

Equality holds if and only if at least one of $X, Y$ is zero almost surely, or there is a constant $c > 0$ with $∣ X ∣^{2} = c ∣ Y ∣^{2} P -almost surely .$

Facts & Assumptions

Given: Real random variables $X, Y \in L^{2} (P)$ .

[L1]

Holder's inequality on a probability space specializes to $E [∣ X Y ∣] \leq ∥ X ∥_{2} ∥ Y ∥_{2}$ (Holder's inequality for random variables).

[L2]

The $L^{2}$ Cauchy-Schwarz equality criterion is already proved for general measure spaces (Cauchy-Schwarz inequality for $L^{2}$ ).

Proof

technique · direct

1.1

Step [L1] at $p = q = 2$ gives $E [∣ X Y ∣] \leq ∥ X ∥_{2} ∥ Y ∥_{2} = (E [X^{2}])^{1 / 2} (E [Y^{2}])^{1 / 2} .$

2.1

The equality clause is exactly the probability-measure specialization of [L2].

step 1.1L2∎

Depends on

Used by

Dependency tree · two levels

5 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

J. R. Norris, Probability and Measure, Theorem 4.4.1 (standard reference, not scraped)