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Finite counting measure recovers finite Holder and implies the signed Cauchy-Schwarz inequality
Via is the space of counting measure, the measure space with counting measure identifies with the published -norm structure of The -norms for rational , and . Under that identification, Holder's inequality for integrals, including the endpoint cases becomes Holder's inequality for finite sums and conjugate real exponents for . At , Cauchy-Schwarz inequality for gives the stronger absolute-product estimate
The real triangle inequality then gives , recovering the signed estimates in The Cauchy-Schwarz inequality for finite sums and Cauchy-Schwarz with its equality case, the triangle inequality for , the parallelogram law and polarisation. Thus the integral and finite forms are compatible, but the absolute-product and signed left sides are not identical.
Depends on
- Cauchy-Schwarz inequality for $L^2$
- Holder's inequality for integrals, including the endpoint cases
- Holder's inequality for finite sums and conjugate real exponents
- The Cauchy-Schwarz inequality for finite sums
- Cauchy-Schwarz $\lvert\langle x,y\rangle\rvert \le \lVert x\rVert_2\lVert y\rVert_2$ with its equality case, the triangle inequality for $\lVert\cdot\rVert_2$, the parallelogram law and polarisation
- The $p$-norms $\lVert x\rVert_p$ for rational $p \ge 1$, and $\lVert x\rVert_\infty$
- $\ell^p$ is the $L^p$ space of counting measure
Used by
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Sources
- Richard L. Wheeden and Antoni Zygmund, Measure and Integral, Chapter 8 (standard reference, not scraped)