Every Vector Space Has a Basis
Theorem Every vector space $V$ has a basis. Proof The case $V$ is zero space is trivial, so we just consider the case $V$ is not zero. We will use “Zorn’s lemma” in the following proof. Let $P$ be the collection of all linearly independent subset of $V$. Note that $P$ is partially ordered with respect to “inclusion”. Also note that $P$ is not empty, since it must contain $\{v\}$ for some $v \in V$. ...