I wonder about the following statement:
Unit ball is compact in a real Norm Linear Space (NLS) iff the space is finite dimensional.
Is this statement true? How would I go about proving this? I don't want to use that every norm in a finite dimensional real NLS is equivalent because I am using the previous result to prove the later.
Best Answer
The unit ball is always closed in a normed (real or complex) linear space. You may be thinking of compactness:
For the first direction, prove the contrapositive by fixing $\epsilon > 0$ and then choosing a countable set of linearly independent vectors with length less than $1$, so that each subsequent vector is further than $\epsilon$ from the span of the previous vectors. This can be done with Riesz's lemma. This is a sequence with no convergent subsequence. For the converse direction, pick a basis and consider $\mathbb{R}^n$.
Thank you for the correction, Martin.