Mathlib theorem · Existing formal mathematics
ProofAtlas

Existing in Mathlib · Linear algebra and analysis

Cauchy–Schwarz Inequality

The amount by which two real or complex inner-product-space vectors align can never exceed the product of their lengths.

Exact theorem

Exact Mathlib statement

theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖

The theorem at a glance

Cauchy–Schwarz inequality at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

Vector alignment is bounded by the product of vector lengths. Explanatory diagram.
Detailed visual description

The poster presents Mathlib's real-or-complex inner-product-space inequality, a projection-based interpretation, and the boundary that this declaration alone does not characterize equality.

Statement structure

From hypotheses to conclusion

Statement map for Cauchy–Schwarz InequalityThe norm of an inner product is at most the product of the two vector norms. The pinned upstream declaration is norm_inner_le_norm. The exact checked statement is theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖.Mathematical readingThe norm of an innerproduct is at most theproduct of the twovector norms.Pinned declarationmathlib ·norm_inner_le_normExact checked formtheoremnorm_inner_le_norm {ᵕ :Type*} {E : Type*}[RCLike ᵕ][SeminormedAddCommGroupE] [InnerProductSpace ᵕE] (x y : E) : ‖inner ᵕx y‖ ≤ ‖x‖ * ‖y‖Statement map for Cauchy–Schwarz InequalityThe norm of an inner product is at most the product of the two vector norms. The pinned upstream declaration is norm_inner_le_norm. The exact checked statement is theorem norm_inner_le_norm {ᵕ : Type*} {E : Type*} [RCLike ᵕ] [SeminormedAddCommGroup E] [InnerProductSpace ᵕ E] (x y : E) : ‖inner ᵕ x y‖ ≤ ‖x‖ * ‖y‖.Mathematical readingThe norm of an innerproduct is at most theproduct of the twovector norms.Pinned declarationmathlib ·norm_inner_le_normExact checked formtheoremnorm_inner_le_norm {ᵕ :Type*} {E : Type*}[RCLike ᵕ][SeminormedAddCommGroupE] [InnerProductSpace ᵕE] (x y : E) : ‖inner ᵕx y‖ ≤ ‖x‖ * ‖y‖

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

The inner-product magnitude cannot exceed the product of the two vector norms. Explanatory scientific diagram.
Detailed visual description

A sparse inner-product-space construction isolates two vectors, the projected component, and a perpendicular remainder without adding an equality condition not present in the selected declaration.

Why it matters

A mathematical landmark

Cauchy–Schwarz is the basic inequality of inner-product geometry. It controls projections, yields the triangle inequality, and supports foundational estimates across linear algebra, analysis, probability, and geometry.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally
Locally reproducedExact upstream declaration replayed
Reviewed pageCurrent public presentation reviewed
Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This target indexes Mathlib's norm inequality for the inner product of two vectors in a real or complex inner-product space. It does not include the equality characterization or claim a new proof.

Source and local evidence

Where the theorem comes from

Existing declaration
norm_inner_le_norm in mathlib
Relationship
The checked artifact is the upstream declaration itself
Local Lean evidence
artifact.library.mathlib.cauchy-schwarz-inequality.v001
Source
Open the pinned upstream reference