I have been trying to solve two questions concerning norm-defining and subspace. This I am trying to do with respect to Banach-spaces. Allow me to introduce the problem (it's a bit long so bear with me):
I am considering a sequence of positive real numbers. We define the weighted -space by:
I want to show that the expression given by:
defines a norm on .
I now consider a special choice:
and wan't to show that is a subspace of .
The following is what I have done so far:
Solution to question 1:
The following three condition must be fulfilled in order for a function to define a norm:
For the sequences and we have:
It is clear that the last inequality is satisfied only if . Furthermore we see that only if . The first requirement is hereby fulfilled.
For a scalar we have that:
Requirement 2 is hereby met.
Having trouble. Assistance needed.
Solution to question 2
Having trouble. Help needed.
Sorry for the long post but I hope that someone can be of assistance regarding showing of c and question 2 . Your help is greatly appreciated.
Thank you very much.