I need to show that the operator defined by has norm 1
There is a hint on the question saying show it must be at least for all
The norm for is
Any help would be great
Okay, so I have shown is linear, pretty straighforward. I am stuggling with the bounded part.
I think I somehow need to show for some . Please correct me if I'm wrong.
I went along the lines of the cauchy schwarz inequality but I'm a bit stuck.
Thanks in advance
If you caculate the integrals for and for the function , you should find that , which may not be "at least " as the hint would like, but it serves the same purpose of being close to 1.
The reason that is that if f, g are functions satisfying , for all x in the interval [a,b], then . In this case, since for , it follows that . Therefore .