[Math] Compact normed vector space

compactnessfunctional-analysis

Let $V$ be a normed vector space.If $V\neq \{0\}$ is it true that our space cannot be compact?

Best Answer

Here is a compact normed vector space. Let $\mathbb F_2$ be the field with two elements $0,1$. Then $V:=\mathbb F_2$ is a $\mathbb F_2$ vector space. With the norm $$ \|v\|:= \begin{cases} 0 & v=0\\ 1 & v=1\end{cases} $$ is becomes a normed vector space. Of course, it is compact.

Seriously, normed vector spaces are usually defined over the reals or complex numbers. Then if $V\ne \{0\}$ there is a non-zero vector $v\in V$. Now consider the sequence $v_n:=n\cdot v$. This sequence does not have a convergent subsequence, hence $V$ cannot be compact.