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--- math105-s22:hw:hw8 [2022/03/12 04:46]
pzhou
+++ math105-s22:hw:hw8 [2026/02/21 14:41] (current)
@@ Line 11: / Line 11: @@
   * (Hölder inequality), for $p,q \geq 1$ that $1/q+1/p=1$, we have
 $$ (\sum_{i=1}^n |x_i y_i|) \leq (\sum_i |x_i|^p)^{1/p} (\sum_i |y_i|^q)^{1/q} $$
-  * (Minkowski inequality) for any $p\geq 1$, $$(\sum_{i=1}^n |x_i + y_i|^p)^{1/p} \leq (\sum_i |x_i|^p)^{1/p} +  (\sum_i |y_i|^q)^{1/q} $$
+  * (Minkowski inequality) for any $p\geq 1$, $$(\sum_{i=1}^n |x_i + y_i|^p)^{1/p} \leq (\sum_i |x_i|^p)^{1/p} +  (\sum_i |y_i|^p)^{1/p} $$
+Read about the proof (in wiki, or any textbook about functional analysis, say Folland). Why it works?

Lecture Notes