---
tag: 00A0RPAQ8A
kind: definition
name: Normed vector space
msc: [46B20]
defines: 03J5RAKB30
uses: [08NYZ2FS6B]
sources:
  - url: https://en.wikipedia.org/wiki/Normed_vector_space
    locator: Definition
---

## 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|0V2WP9B1MF]] $\norm{\cdot}$ 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|0QVNCJXRY3]] 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.