normed algebra

A ring A is said to be a normed ring if A possesses a norm , that is, a non-negative real-valued function :A such that for any a,bA,

  1. 1.

    a=0 iff a=0,

  2. 2.


  3. 3.

    -a=a, and

  4. 4.



  • If A contains the multiplicative identityPlanetmathPlanetmath 1, then 0<111 and so 11.

  • However, it is usually required that in a normed ring, 1=1.

  • defines a metric d on A given by d(a,b)=a-b, so that A with d is a metric space and one can set up a topologyMathworldPlanetmath on A by defining its subbasis a collection of B(a,r):={xAd(a,x)<r} called open balls for any aA and r>0. With this definition, it is easy to see that is continuous.

  • Given a sequence {an} of elements in A, we say that a is a limit pointPlanetmathPlanetmath of {an}, if


    By the triangle inequalityPlanetmathPlanetmath, a, if it exists, is unique, and so we also write

  • In addition, the last condition ensures that the ring multiplication is continuous.

An algebra A over a field k is said to be a normed algebra if

  1. 1.

    A is a normed ring with norm ,

  2. 2.

    k is equipped with a valuationMathworldPlanetmath ||, and

  3. 3.

    αa=|α|a for any αk and aA.


  • Alternatively, a normed algebra A can be defined as a normed vector spacePlanetmathPlanetmath with a multiplication defined on A such that multiplication is continuous with respect to the norm .

  • Typically, k is either the reals or the complex numbersMathworldPlanetmathPlanetmath , and A is called a real normed algebra or a complex normed algebra correspondingly.

  • A normed algebra that is completePlanetmathPlanetmathPlanetmathPlanetmath with respect to the norm is called Banach algebraMathworldPlanetmath (the underlying field must be complete and algebraically closedMathworldPlanetmath), paralleling with the analogy with a Banach spaceMathworldPlanetmath versus a normed vector space.

  • Normed rings and normed algebras are special cases of the more general notions of a topological ring and a topological algebra, the latter of which is defined as a topological ring over a field such that the scalar multiplication is continuous.


  • 1 M. A. Naimark: Normed Rings, Noordhoff, (1959).
  • 2 C. E. Rickart: General Theory of Banach Algebras, Van Nostrand, 1960.
Title normed algebra
Canonical name NormedAlgebra
Date of creation 2013-03-22 16:11:38
Last modified on 2013-03-22 16:11:38
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 13
Author CWoo (3771)
Entry type Definition
Classification msc 46H05
Related topic GelfandTornheimTheorem
Related topic SuperfieldsSuperspace
Defines normed ring
Defines topological algebra
Defines real normed algebra
Defines complex normed algebra