Analysis MOC

Hölder’s inequality

Let be a measure space and be Hölder conjugate, i.e. . Then for any measurable functions ¹ anal

where denotes the [[Lebesgue space|-norm]].

Proof

Let and . If or the inequality holds trivially, and similarly if or .

Now assume , and take . Taking Young’s inequality with and gives

whence

wherefore

The only case yet to be handled is . It follows immediately that almost everywhere and thus , as required.

The Cauchy-Schwarz inequality for the [[Lebesgue space|-norm]] is the case .

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Footnotes

1. If in addition, if and are finite, , and , then iff are linearly dependent. ↩