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 [ilmath]F[/ilmath], [ilmath](X,F)[/ilmath]
  • where [ilmath]F[/ilmath] is either [ilmath]\mathbb{R} [/ilmath] or [ilmath]\mathbb{C} [/ilmath]
• Equipped with a norm, [ilmath]\Vert\cdot\Vert[/ilmath]

We denote such a space by:

• [ilmath](X,\Vert\cdot\Vert,F)[/ilmath] or simply [ilmath](X,\Vert\cdot\Vert)[/ilmath] if the field is obvious from the context.

Names

A normed space may also be called:

• Normed linear space (or n.l.s)