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--- math105-s22:hw:hw8 [2022/03/11 20:46]
pzhou created
+++ math105-s22:hw:hw8 [2022/03/11 20:50] (current)
pzhou
@@ Line 8: / Line 8: @@
   * optional: if we use  $\| - \|_{max}$  norm on  $\R^n$ , how to compute the operator norm  $\|T\|$ ?
-. Read about Hölder inequality and Minkowski inequality. Can you come up with an elementary prove for
+. Read about Hölder inequality and Minkowski inequality. In the simplest setting, we have
-  * (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} $
+  * (Hölder inequality), for  $p,q \geq 1$  that $1/q+1/p=1$, we have
-  * (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} $
+$$ (\sum_{i=1}^n |x_i y_i|) \leq (\sum_i |x_i|^p)^{1/p} (\sum_i |y_i|^q)^{1/q} $$
+  * (Minkowski inequality) for any  $p\geq 1$ , $$(\sum_{i=1}^n |x_i + y_i|^p)^{1/p} \leq (\sum_i |x_i|^p)^{1/p} +  (\sum_i |y_i|^p)^{1/p} $$
+Read about the proof (in wiki, or any textbook about functional analysis, say Folland). Why it works?

Lecture Notes