Any infinite dimensional vector space has a basis?

Question

How should I prove that any infinite dimensional vector space has a basis?
I have heard that there is a proof using Zorn's Lemma but I do not know how. In addition, are there different approaches/proofs to this problem?

Answer 1

Let $V$ be a vector space, and let $\Sigma$ be the collection of all linearly independent subsets of $V$, partially ordered by inclusion. Let $C$ be a chain in $\Sigma$, and consider $T = \bigcup_{S \in C} S$. Now $T$ is a linearly independent subset of $V$ (otherwise, a nontrivial linear combination of finitely many elements in $T$ is zero, but these finitely many elements all belong to some $S \in C$–a contradiction). Thus, every chain in $\Sigma$ has an upper bound in $\Sigma$, so $\Sigma$ has maximal elements, by Zorn's Lemma.

Let $T$ be a maximal linearly independent subset of $V$. We will show that $T$ is a basis. Suppose there exists $v \in V$ which is not a linear combination of elements of $T$. Then $v$ can be added to $T$ to create a larger linearly independent set, but this contradicts the maximality of $T$. Thus, every element of $V$ is a linear combination of element of $T$, and so $T$ is a basis.

Edit. In fact, the existence of a basis in every vector space is equivalent to Zorn's Lemma, so any proof of this result will somehow rely on Zorn's Lemma, or one of its many equivalent forms.