proof of Hölder inequality
First we prove the more general form (in measure spaces).
Let be a measure space and let , where and .
The case and is obvious since
Also if or the result is obvious. Otherwise notice that (applying http://planetmath.org/node/YoungInequalityYoung inequality) we have
hence the desired inequality holds
If and are vectors in or vectors in and -spaces we can specialize the previous result by choosing to be the counting measure on .
In this case the proof can also be rewritten, without using measure theory, as follows. If we define
|Title||proof of Hölder inequality|
|Date of creation||2013-03-22 13:31:16|
|Last modified on||2013-03-22 13:31:16|
|Last modified by||paolini (1187)|
|Synonym||proof of Hölder inequality|
|Synonym||proof of Holder’s inequality|