Basis and dimension

Let $X$ be a vector space over $K$, and let $B\subset X$.

A set $B$ is a basis (also called a Hamel basis) of $X$ if:

1. $B$ is linearly independent , and
2. every $x\in X$ can be written as a finite linear combination of elements of $B$, i.e., there exist $m\in\mathbb{N}$, $x_1,\dots,x_m\in B$, and $\alpha_1,\dots,\alpha_m\in K$ such that $x=\sum_{i=1}^m \alpha_i x_i.$

If $X$ has a basis with finitely many elements, then $X$ is finite-dimensional, and the number of basis vectors is the dimension $\dim(X)$. If no finite basis exists, $X$ is infinite-dimensional (often written $\dim(X)=\infty$).

The existence of a Hamel basis in full generality uses set-theoretic choice; see extension to a basis .

Examples:

• The set $\{e_1,\dots,e_n\}$ is a basis of $\mathbb{R}^n$, so $\dim(\mathbb{R}^n)=n$.
• The polynomials of degree $\le m$ form a vector space with basis $\{1,t,\dots,t^m\}$, so its dimension is $m+1$.
• The space of all sequences $s$ is infinite-dimensional (no finite set can generate all sequences by finite linear combinations).