Normed vector space

A normed vector space is a vector space $V$ equipped with a norm $\|\cdot\|$ on $V$.

The norm induces a metric space structure via $d(u,v)=\|u-v\|$, so notions from topology (like continuity and convergence) apply. If a normed vector space is complete with respect to this metric, it is a Banach space . Norms on linear maps are captured by the operator norm .

Examples:

• $(\mathbb{R}^n,\|\cdot\|_2)$ is a normed vector space.
• The space $C([0,1])$ of continuous real-valued functions on $[0,1]$ with the sup norm is a normed vector space.
• Any finite-dimensional vector space over $\mathbb{R}$ or $\mathbb{C}$ with any norm is a normed vector space.