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Cauchy–Schwarz: , with equality exactly for linearly dependent vectors
Statement
For vectors in a real or complex inner product space,
Equality holds if and only if and are linearly dependent, including the case in which either vector is zero.
Facts & Assumptions
Given: Vectors in an inner product space over or .
The inner product is linear in the first variable, conjugate-linear in the second, conjugate symmetric, and positive definite (Real and complex inner product spaces, with the inner product linear in the first argument).
The norm is the nonnegative square root of the diagonal pairing (The norm induced by a real or complex inner product, Existence and uniqueness of -th roots: a unique with ).
Complex modulus satisfies and vanishes exactly at zero (Conjugation is an involutive real-field automorphism, , and modulus is definite, multiplicative, and subadditive).
A two-vector list is dependent exactly when a nontrivial scalar combination vanishes (Linear independence: a finite list is independent when forces every , and a subset is independent when every injective finite list into is independent).
Proof
If , both sides are zero and the pair is dependent. Suppose , put , and use [L1] to expand .
Conversely, if are dependent and neither is zero, write ; then . If either is zero, equality is immediate.
Multiplying step 1.1 by the positive number gives . Since both sides of the desired inequality are nonnegative, factoring the difference of their squares gives the stated inequality.
Under , equality in step 2.1 holds exactly when , which by [L1] is exactly . Thus equality implies dependence. The already separated case does too.
Steps 2.1, 3.1, and 1.2 prove the inequality and both equality directions.
Depends on
- Real and complex inner product spaces, with the inner product linear in the first argument
- The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product
- Conjugation is an involutive real-field automorphism, $z\overline z=|z|^2$, and modulus is definite, multiplicative, and subadditive
- Existence and uniqueness of $n$-th roots: a unique $a^{1/n} \ge 0$ with $(a^{1/n})^n = a$
- Linear independence: a finite list $v : n \to V$ is independent when $\sum_{i<n} \lambda_i v_i = 0_V$ forces every $\lambda_i = 0_F$, and a subset $S \subseteq V$ is independent when every injective finite list into $S$ is independent
Used by
- The inner-product norm is definite, homogeneous, and satisfies the triangle inequality Corollary
- Hermitian metric and L² pairing on a compact Riemann surface Definition
- Lipschitz polyhedral boundary, orientation, and trace spaces Definition
- Pointwise norm and angle from a riemannian metric Definition
- Schwartz/distribution spaces and Hilbert Fock conventions Definition
- Bounded linear predictors have a radius-dependent bound Example
- Norm-attaining functionals on a Hilbert space Example
- Path determinants dₖ=dₖ₋₁-cos²(π/m)dₖ₋₂ and the three-arm inequality Example
- The Gram determinant of two vectors is ‖ u‖²‖ v‖²-|⟨ u,v⟩|² and detects dependence Example
- A complete no-return extension of a strictly convex manifold Lemma
- A computed unitary radix-two butterfly has error at most (10u + εtw) times its input norm Lemma
- A proper matrix-product cover from the mixed (2,1) norm Lemma
- A sphere net controls a real or complex Hermitian operator norm Lemma
- A truncated covering integral bounds empirical Rademacher complexity Lemma
- Cartan-number products, allowed edge labels, tree scalings and reflection stability Lemma
- Certified action bounds control one exponential-integrator step Lemma
- Cheeger sweep and layer cake Lemma
- Chevalley basis and real structure constants Lemma
- Class-collision loss is bounded by within-class covariance deviation Lemma
- Class-conditional sub-Gaussian concentration yields a contrastive margin bound Lemma
- Compression dimension bounds entanglement fidelity Lemma
- Dolbeault green operator is compact on the orthogonal complement of the kernel Lemma
- Dominant projection and strict parabolic weight decrease Lemma
- Exclusions for positive definite diagrams: trees, valency, labels, chains and arms Lemma
- Expander walk contraction Lemma
- Finite channel information maxima are attained Lemma
- Finite torus fourier orthogonality and affine change Lemma
- Finite unitary averaging and a decoupling second-moment bound Lemma
- Gaussian sample covariance converges in operator norm when dimension is little-o of sample size Lemma
- Gentle measurement bound in finite dimensions Lemma
- Hilbert tensor construction: inner product, completion and product bases Lemma
- Intersection of root subcomplexes and purity under convexity Lemma
- Normalized Hermitian inequality SDPs have equal primal and dual values Lemma
- Overlap control gives a union lower bound Lemma
- Partial trace contracts the trace norm on Hermitian operators Lemma
- Projective Chord Distance Formula Lemma
- Pure-state trace distance is bounded by vector error Lemma
- Rayleigh energy perturbation for normalized finite states Lemma
- Robust one-step OAA including ancilla leakage Lemma
- Six-operation complex multiplication has a normwise roundoff bound Lemma
…and 34 more results.
Dependency tree · two levels
35 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)
- Sergei Treil, Linear Algebra Done Wrong, Ch. 5, §5.1 (standard reference, not scraped)