# Definition:Normed Vector Space

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## Contents

## Definition

Let $\struct {K, +, \circ}$ be a normed division ring.

Let $V$ be a vector space over $K$.

Let $\norm {\,\cdot\,}$ be a norm on $V$.

Then $\struct {V, \norm {\,\cdot\,} }$ is a **normed vector space**.

## Also known as

When $\norm {\,\cdot\,}$ is arbitrary or not directly relevant, it is usual to denote a **normed vector space** merely by the symbol $V$.

A **normed vector space** is also known as a **normed linear space**.

## Also see

- Results about
**normed vector spaces**can be found here.

## Sources

- 1975: W.A. Sutherland:
*Introduction to Metric and Topological Spaces*... (previous) ... (next): $2.2$: Examples: Remark - 2013: Francis Clarke:
*Functional Analysis, Calculus of Variations and Optimal Control*: $1.1$: Basic Definitions - 2017: Amol Sasane:
*A Friendly Approach to Functional Analysis*... (previous) ... (next): $\S 1.2$: Normed and Banach spaces. Normed spaces