Prove that V is infinite-dimensional if and only if there is a sequence v1,v2,… of vectors in V such that v1,…,vm is linearly independent for every positive integer m.
I already know that whether a vector space is finite-dimensional depends on whether there exist some list of vectors in it spans it, but I cannot give a rigorous proof on the problem above, please help me, thanks!
Best Answer
Note: I assume the axiom of choice, so that every vector space has a basis. Otherwise, I'm not sure how we should define dimension.
Suppose that $V$ is finite dimensional. Let $n$ be the dimension of $V$. Then $v_1,\dots,v_m$ cannot be linearly independent if $m > n$.
For the converse: if $V$ is infinite dimensional, then $V$ has an infinite basis. Any countable subset of this basis satisfies the hypothesis on $\{v_1,v_2,\dots\}$.