Show that if u,v \in RR^m, then |u * v| <= norm(u)_(oo) norm(v)_1

odigavz

odigavz

Answered question

2022-08-11

Show that if u , v R m , then | u v | | | u | | | | v | | 1
By Cauchy-Schwarz,
| u v | | | u | | 2 | | v | | 2
Note also that | | u | | 1 m | | u | | 2 m | | u | | , and | | v | | | | v | | 2 | | v | | 1 , so we get
| u v | m | | u | | | | v | | 1
How to get rid of m to get the desired inequality?

Answer & Explanation

Ashlynn Stephens

Ashlynn Stephens

Beginner2022-08-12Added 25 answers

Let u = ( u 1 , , u m ) and v = ( v 1 , , v m ). You have
| u v | = | k = 1 m u k v k | k = 1 m | u k v k | u k = 1 m | v k | = u v 1
(you can put u out of the sum because all the | u k | are less than or equal to u )

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