Dense subset

The concept of denseness is at the center of approximation theory. That is, a space $$W \subseteq V$$ may not contain all elements of $$V$$, but if $$W$$ is dense in $$V$$, then any element of $$V$$ can be approximated arbitrarily well by elements of $$W$$.

Formally, a subset $$W$$ of a normed vector space $$V$$ is said to be dense in $$V$$ if for each $$\mathbf{v} \in V$$ and each $$\epsilon > 0$$ there exists $$\mathbf{w}\in W$$ such that (Def. 2.3.1 ) $$||\mathbf{w-v}||\le \epsilon$$.