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1. Holder's Inequality and Minkowski's inequality. We fix p, q [1, ] such that
 

Summary: 1. H¨older's Inequality and Minkowski's inequality.
We fix p, q [1, ] such that
1
p
+
1
q
= 1.
Definition 1.1. Suppose f : Rn
R is Lebesgue measurable. We let
||f||p = |f(x)|p
dx
1/p
if p <
and we let
||f|| = sup{t (0, ) : Ln
({|f| > t}) > 0}.
Proposition 1.1. Suppose f : Rn
R is Lebesgue measurable and c R. Then
||cf||p = |c|||f||p.

  

Source: Allard, William K. - Department of Mathematics, Duke University

 

Collections: Mathematics