Finite-Dimensional Vector Space

Definition: Finite-dimensional vector space

A vector space is called finite-dimensional if some list of vectors in it spans the space.

Definition: Infinite-dimensional vector space

A vector space is called infinite-dimensional if it is not finite-dimensional.

Definition: Finite-dimensional subspaces

Every subspace of a finite-dimensional vector space is finite-dimensional.

Proof. Suppose $V$ is finite-dimensional and $U$ is a subspace of $V$. We need to prove that $U$ is finite-dimensional. We do this through the following multistep construction.

Step 1: If $U=\{0\}$, then $U$ is finite-dimensional and we are done. If $U\ne \{0\}$, then choose a nonzero vector $u_1\in U$.

Step $k$: If $U=\text{span}(u_1,\dots ,u_{k-1})$, then $U$ is finite-dimensional and we are done. If $U\ne \text{span}(u_1,\dots ,u_{k-1})$, then choose a vector $u_k\in U$ such that

$$u_k\notin \text{span}(u_1,\dots ,u_{k-1})$$