I doubt that I misunderstand this part:
please clarify me:
if $V$ is a finite-dimensional vector space and we have an independent set like $S$ then we can expand $S$ to a basis for the vector space easily.
if $V$ is an infinite-dimensional vector space and we have an independent set $S$ then again we can expand $S$ to a basis for the vector space but this time the procedure is more complicated and for proving this we must use Zorn's lemma
are this two statements correct?
Best Answer
@mathnoob Let $\mathcal{P}=\{L\supset S|L\text{ is linearly independent}\}$.This is nonempty since $S\in \mathcal{P}$. Then $(\mathcal{P},\subset)$ is a poset with set inclusion. Use zorn's lemma on this to get a maximal linearly independent set $B$ containing $S$. Then $B$ is an extension of $S$.