Normed space

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


A normed space is a[1][2]:

  • 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.


A normed space may also be called:

  • Normed linear space[1] (or n.l.s)


  1. 1.0 1.1 Functional Analysis - George Bachman and Lawrence Narici
  2. Analysis - Part 1: Elements - Krzysztof Maurin