Normed space

See: Subtypes of topological spaces for a discussion of relationships of normed spaces.

Definition

A normed space is a:

• vector space over the field $F$, $(X,F)$
  • where $F$ is either $\mathbb{R}$ or $\mathbb{C}$
• Equipped with a norm, $\Vert\cdot\Vert$

We denote such a space by:

• $(X,\Vert\cdot\Vert,F)$ or simply $(X,\Vert\cdot\Vert)$ if the field is obvious from the context.

Names

A normed space may also be called:

• Normed linear space (or n.l.s)