Originally Posted by

**macrone** 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.

||z-y||>= theta

My question:

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.

a=inf||v-y||

since Y is compact there exists y_o in Y such that

a=||v-y_o||

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||?