I would like to show that the sequence (the k'th entry is 1) is an orthonormal basis for the space
I know that to show this I must:
1) Show that is a basis
2) Show that
3) Show that
My questions is only concerning point 1): I have already shown that is a basis for . Can the sequence be a basis for both spaces? If yes, would it be acceptable to use the method used in the case of ?
The fact that it is a basis does not depend on the norm of the space. Think about it, what part of involves a norm?
All of the little L spaces are the same vector space, so once you have a basis, it is a basis for all of them.