Is the statement also true in infinite dimensional vector spaces?

Question

I am trying to prove that:

"A basis is equivalent to a maximal linearly independent set"

I am wondering if this statement is true for infinite dimensional vector spaces.

Can someone help me answer this question please?

Answer 1

If you define a basis as a linearly independent set which spans the entire space, then the answer is yes.

Suppose $B$ is a basis for a vector space $V$. Let $B \subseteq B' \subseteq V$ be arbitrary. Then for any $b \in B'\setminus B$ by definition of a basis we have $b \in \operatorname{span} B$ so $B'$ cannot be linearly independent. Therefore $B$ is a maximal linearly independent set in $V$.

Conversely, let $B\subseteq V$ be a maximal linearly independent set in $V$. Let $x \in V$ be arbitrary. If $x \notin \operatorname{span} B$, then $B\cup \{x\}$ would be a linearly independent set containing $B$, contradicting the maximality of $B$. Therefore $x \in \operatorname{span} B$. Since $x \in V$ was arbitrary, we conclude that $\operatorname{span} B = V$ so $B$ is a basis for $V$.