I was given the very first inf definition, and have to prove that 3rd is the same as the given one. What I tried is taking sup on both sides of llAvll =< llAll llvll, but failed. Any help would be greatly appreciated.
Thanks
I was given the very first inf definition, and have to prove that 3rd is the same as the given one. What I tried is taking sup on both sides of llAvll =< llAll llvll, but failed. Any help would be greatly appreciated.
Thanks
Let and let .
Consider any such that for all and choose any with .
Then and so , i.e. . Taking the inf over all such , we deduce that .
If is non-zero, let . Then and so .
It follows by linearity that for all non-zero and therefore for all , since it is clearly true when .
Hence for all when . By the definition of , we see that .
Thus we have .