With Riesz's lemma, I know that if i let X be a normed space. and Y and Z be subspaces of a normed space X then if Y is closed proper subset of Z then for every real number theta on interval (0,1) there exist a ||z||=1 s.t.
When Y is finite dimensional then theta can be (0,1]
i am having trouble proofing this.
I believe that i want to show that since Y is bounded and closed it is compact. Therefore
their exists value v in Z-Y s.t.
since Y is compact there exists y_o in Y such that
and if this is true i could easily finish the proof from there.
Basically my two questions are:
How do i Show that Y is bounded. Is it automatically implied because it is finite dimensional and closed?
If Y is compact can i assume that there exists a y_0 in Y s.t. a=||v-y_o||?