TheoremStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-22

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Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs

Statement

For all vectors $x, y$ in a real or complex inner-product space,

$∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥,$

with equality if and only if $x$ and $y$ are linearly dependent.

Facts & Assumptions

[A1]

The pairing is linear in the first argument, conjugate-linear in the second, conjugate symmetric and positive definite, and the induced length satisfies $⟨ v, v ⟩ = ∥ v ∥^{2}$ with $∥ v ∥ \geq 0$ and $∥ v ∥ = 0$ exactly for $v = 0$ (Real and complex inner-product spaces and their induced length).

[A2]

The induced length is the unique nonnegative square root of the diagonal pairing, and positive definiteness makes the radicand a nonnegative real (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product).

[A3]

Every nonnegative real has a unique nonnegative square root: there is exactly one $s \geq 0$ with $s^{n} = a$ for each $n \geq 1$ (Existence and uniqueness of $n$ -th roots: a unique $a^{1 / n} \geq 0$ with $(a^{1 / n})^{n} = a$ ).

[A4]

For complex scalars $z \overline{z} = ∣ z ∣^{2}$ , $∣ z ∣ \geq 0$ , and $∣ z ∣ = 0$ exactly for $z = 0$ (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive).

[A5]

For real scalars $∣ x ∣ \geq 0$ , $∣ x ∣ = 0$ exactly for $x = 0$ , and $∣ x y ∣ = ∣ x ∣ ∣ y ∣$ (Basic properties of the absolute value).

[A6]

For nonnegative reals, $a \leq b$ if and only if $a^{2} \leq b^{2}$ (Squaring is monotone on the nonnegatives).

[A7]

A finite list is linearly dependent when some choice of scalars, not all zero, makes the corresponding combination vanish; for the two-element list $(x, y)$ this means that $a x + b y = 0$ for scalars $a, b$ not both zero (Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i < n} λ_{i} v_{i} = 0_{V}$ forces every $λ_{i} = 0_{F}$ , and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent).

Proof

technique · direct

Given: Vectors $x, y$ in a real or complex inner-product space $V$ . In the real case read conjugation as the identity and $∣ \cdot ∣$ as the absolute value of $R$ , so that [A4] is replaced by [A5].

1.1

If $y = 0$ , then $⟨ x, y ⟩ = ⟨ x, 0 \cdot y ⟩ = 0 \cdot ⟨ x, y ⟩ = 0$ by conjugate-linearity in the second argument, and $∥ y ∥ = 0$ ; hence $∣ ⟨ x, y ⟩ ∣ = 0 = ∥ x ∥ ∥ y ∥$ , while $x, y$ are dependent with witness scalars $(a, b) = (0, 1)$ because $0 \cdot x + 1 \cdot 0 = 0$ and $b \neq 0$ .

A1A2A7

1.2

Suppose now $y \neq 0$ and set $c = ⟨ x, y ⟩ / ⟨ y, y ⟩$ , a well-defined scalar because $⟨ y, y ⟩ = ∥ y ∥^{2} > 0$ ; expanding with linearity in the first argument, conjugate-linearity in the second and conjugate symmetry gives $0 \leq ∥ x - c y ∥^{2} = ∥ x ∥^{2} - c \overline{⟨ x, y ⟩} - \overline{c} ⟨ x, y ⟩ + ∣ c ∣^{2} ∥ y ∥^{2} = ∥ x ∥^{2} - ∣ ⟨ x, y ⟩ ∣^{2} / ∥ y ∥^{2}$ .

A1A2A3A4A5algebra

2.1

Multiplying step 1.2 by the positive number $∥ y ∥^{2}$ gives $∣ ⟨ x, y ⟩ ∣^{2} \leq ∥ x ∥^{2} ∥ y ∥^{2}$ , and since $∣ ⟨ x, y ⟩ ∣$ , $∥ x ∥$ and $∥ y ∥$ are nonnegative, monotonicity of squaring on the nonnegatives gives the inequality $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ .

A2A6step 1.2algebra

2.2

If $x, y$ are dependent, then either $y = 0$ , which is step 1.1, or $x = c y$ for some scalar $c$ with $y \neq 0$ ; in the second case $⟨ x, y ⟩ = c ∥ y ∥^{2}$ and $∥ c y ∥^{2} = ⟨ c y, c y ⟩ = ∣ c ∣^{2} ∥ y ∥^{2}$ with both norms nonnegative, so $∥ x ∥ = ∣ c ∣ ∥ y ∥$ by uniqueness of nonnegative square roots, and $∣ ⟨ x, y ⟩ ∣ = ∣ c ∣ ∥ y ∥^{2} = ∥ x ∥ ∥ y ∥$ .

A1A2A3A4A5A6A7step 1.1algebra

3.1

Conversely suppose $∣ ⟨ x, y ⟩ ∣ = ∥ x ∥ ∥ y ∥$ and $y \neq 0$ ; then $∥ y ∥^{2} > 0$ and step 1.2 gives $∥ x - c y ∥^{2} = ∥ x ∥^{2} - ∥ x ∥^{2} = 0$ , so $x - c y = 0$ by positive definiteness, that is $x = c y$ and the pair is dependent by [A7]; together with steps 1.1 and 2.2 this proves the inequality and both directions of the equality statement.

A1A7step 1.1step 1.2step 2.2∎

Depends on

Used by

A separable infinite-dimensional Hilbert space is ℓ² Corollary
The induced length is a norm Corollary
An inner-product space need not be complete Counterexample
Numerical range and numerical radius Definition
The Hilbert-space adjoint of a bounded operator Definition
Trace of a trace class operator Definition
Adjoint, norm and trace of an operator of rank at most one Example
Integral operator trace under a valid diagonal hypothesis Example
Volterra operator is Hilbert Schmidt and quasinilpotent Example
Laplace resolvents of a unitary group Lemma
Norm of a self adjoint operator from its quadratic form Lemma
Nuclear series characterizes trace norm Lemma
Orthogonal complements are closed Lemma
Scalar and complex measures from a pvm Lemma
Spectrum of a positive operator is nonnegative Lemma
Square-summable orthogonal families have norm-convergent finite sums Lemma
The generator of a unitary group is closed and skew-adjoint Lemma
The inner product is jointly continuous Lemma
Weak and strong additivity of orthogonal projections Lemma
A Hilbert space with a given orthonormal basis is ℓ² of the index set Theorem
Cyclicity of the trace Theorem
Existence of a maximal orthonormal family, and maximality as completeness Theorem
Fourier expansion in a Hilbert space Theorem
Hilbert Schmidt operators form a two sided ideal Theorem
Hilbert-adjoint identities Theorem
Min-max principle below the essential spectrum Theorem
Numerical radius is an equivalent operator norm Theorem
Parseval equivalences for an orthonormal family Theorem
Riesz representation for Hilbert spaces Theorem
The norm completion of an inner-product space is a Hilbert space Theorem
Trace class iff product of two Hilbert Schmidt operators Theorem
Trace is absolutely convergent and basis independent Theorem
Von Neumann parameterization of self-adjoint extensions Theorem
Weyl criterion for the essential spectrum Theorem

Dependency tree · two levels

38 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

Theo Bühler and Dietmar Salamon, Functional Analysis, §1.3.3, Lemma 1.40, p.38 (standard reference, not scraped)
Andrew Lin and Casey Rodriguez, MIT 18.102 Introduction to Functional Analysis, Lectures 15–16 (standard reference, not scraped)

Cauchy–Schwarz: ∣⟨x,y⟩∣≤∥x∥ ∥y∥∣⟨x,y⟩∣≤∥x∥∥y∥, with equality exactly for dependent pairs