Normed vector space - Knowlpedia

Empty set and whole space: ∅ , X ∈ T \varnothing,X\in\mathcal T ∅ , X ∈ T .
Arbitrary unions: the union of any collection of members of T \mathcal T T belongs to T \mathcal T T .
Finite intersections: the intersection of finitely many members of T \mathcal T T belongs to T \mathcal T T .

A normed vector space is a vector space over $\mathbb{R}$ or $\mathbb{C}$ , equipped with a norm $\|\cdot\|$ on $V$ .

Remarks

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

For example, on $\mathbb{R}^2$ the Euclidean, $\ell^1$ , and max norms give different lengths but the same notion of convergence. In infinite dimensions, different norms on the same vector space need not define the same topology.

$(\mathbb{R}^n,\|\cdot\|_2)$ 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.