Definition:Normed Vector Space

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 the norm $\norm {\,\cdot\,}$ 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.

Some sources refer just to a normed space.

Also see

• Definition:Banach Space
• Definition:Norm on Vector Space
• Definition:Normed Division Ring

• Results about normed vector spaces can be found here.

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.2$: Examples: Remark
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): normed space
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): norm: 1. (of a vector space)
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): normed space
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): norm: 1. (of a vector space)
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): normed space
• 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
• 2020: James C. Robinson: Introduction to Functional Analysis ... (previous) ... (next) $3.1$: Norms