Normed vector space

Definition

Let $\mathbb{K}$ be $\R$ or $\C$. A normed vector space over $\mathbb{K}$ is a vector space $V$ over $\mathbb{K}$ together with a norm $\norm{\cdot}$ on $V$.

Remarks

A normed space is a metric space with the distance $d(x, y) = \norm{x - y}$. A normed space that is complete for this metric is a Banach space. Each inner product space is a normed space with the induced norm. On a finite-dimensional space, all norms are equivalent, so they define the same topology.

Explanations

L1 human

A normed space is a vector space where each vector has a length. With lengths, we also have distances, and so limits and continuity. This is the setting of much of analysis.