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Cauchy-Schwarz inequality for
Statement
If , then
Equality holds if and only if at least one of is zero almost everywhere, or there is a constant with
Facts & Assumptions
Given: Functions .
Holder's inequality holds for conjugate exponents (Holder's inequality for integrals, including the endpoint cases).
The strict-exponent equality criterion for Holder has already been proved (Equality in Holder's inequality for ).
Proof
The exponent is conjugate to itself, so [L1] with gives [L1]
Because , the equality clause is exactly the specialization of [L2] to [L2, step 1.1] .
Depends on
Used by
- Cauchy-Schwarz for random variables Corollary
- Conditional cauchy schwarz inequality Corollary
- Direct integral of a measurable Hilbert field Definition
- Convolution on a discrete group Example
- Extremal length of a rectangle and of a round annulus by hand Example
- Projection onto the constants is the mean Example
- Backward uniqueness for the heat equation on a bounded interval Lemma
- Bounded H1 functionals have compatible local L2 representatives Lemma
- Circular dilatation, quasisymmetry and the analytic definition Lemma
- Compact-open neighbourhoods on the dual give a neighbourhood basis on the group Lemma
- Compactly supported nonnegative transform bumps on the dual Lemma
- Conditional variance is well-defined and has the second-moment formula Lemma
- Continuous characters separate points of an LCA group Lemma
- Finite positive circle measures admit a Lebesgue decomposition under countable choice Lemma
- L² with the integral pairing is a Hilbert space Lemma
- L2-normalised H1 atoms have uniformly bounded H1 norm Lemma
- Parseval pairing on the integrable core Lemma
- Riemann maps of Jordan domains extend to homeomorphisms of the closures Lemma
- The dual representative has uniformly bounded BMO oscillation Lemma
- The Plancherel transform range is dense in L² of the dual Lemma
- Mixing correlations extend to L2 functions Proposition
- Finite counting measure recovers finite Holder and implies the signed Cauchy-Schwarz inequality Remark
- Calderon-Zygmund operators map L-infinity to BMO Theorem
- Conditional expectation is the l2 orthogonal projection Theorem
- Energy continuous dependence for the forced wave equation Theorem
- Extremal length of the rectangle and of the round annulus Theorem
- Plancherel isometric extension on LCA groups Theorem
- Quadratic covariation of Brownian Ito processes Theorem
- The geometric and analytic definitions of quasiconformality agree Theorem
- The measurable Riemann mapping theorem on the sphere Theorem
Dependency tree · two levels
13 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, Measure, Integration & Real Analysis, Section 7A (standard reference, not scraped)
- John K. Hunter, Measure Theory, Section 7.2 (standard reference, not scraped)