If then . This is trivial if . Now let and . Then
If is finite and , then . This is easy if , because almost everywhere, so . Now let , thus
Finally, we prove an interesting property for -norms: if is a finite measure space, then for any measurable function on the equality holds. We have already seen that . Now for any define , and . Since , we have . Now we take on the left and on the right side: . Taking gives .
|Date of creation||2013-03-22 15:22:05|
|Last modified on||2013-03-22 15:22:05|
|Last modified by||yark (2760)|