The key thing that you need to notice is that every linear functional on is of the form , for some scalars . All you have to do then is to take .
Hi All,
Could you please help with the following?
Let X be a normed vector space with Banach dual space X'. Let be in X' such that . Show that
is a well-defined linear map on a subspace of . Extending T to all of , show that there are such that
Conclude that, for any finite-dimensional subspace M of X',
Now, by definition, for
Now, I can show that T is well-defined and linear on the subspace .
By the Hahn-Banach theorem, I can extend this to a function such that and . But where do I go from here?
Thanks a lot for this!
Could you please check I've got the last part, ( ), right?
So, want to show Want to show ,
i.e. that (by defn of )
i.e. (by defn of )
which is true, as we started with
Similarly for the other direction. What I don't get - where did we use the bit? And where in this last part did we use any of the previous bits of the question?
Thanks in advance!
The place where you needed was right at the beginning of the question, in order to show that is well defined. If the definition is to make sense, it must be true that depends only on . So you need to be sure that if then . But that is equivalent to saying that if for i=1,2,...,n, then . That in turn is equivalent to the condition .
For the last part of the question, you are given a finite-dimensional subspace . Let be a basis for . Then
and
But it follows from the earlier part of the question that that last condition implies that is a linear combination of , which says that . Therefore . You have shown how to prove the reverse inclusion , and so .