Namespaces
Variants
Actions

Difference between revisions of "Normed algebra"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
Line 1: Line 1:
An [[Algebra|algebra]] over the field of real or complex numbers that is at the same time a [[Normed space|normed space]] in which multiplication satisfies some continuity condition. The simplest such condition is separate continuity. Generally speaking, this is weaker than joint continuity in the factors. For example, if one defines the algebraic operations on the set of all finite sequences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067770/n0677701.png" /> coordinate-wise and the norm by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067770/n0677702.png" />, then there arises an algebra in which the multiplication is separately, but not jointly, continuous. The joint continuity of the multiplication in a normed algebra is equivalent to the existence of a constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067770/n0677703.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n067/n067770/n0677704.png" />. In this and only in this case the completion has the structure of a normed algebra that extends the original one and is a [[Banach algebra|Banach algebra]].
+
<!--
 +
n0677701.png
 +
$#A+1 = 4 n = 0
 +
$#C+1 = 4 : ~/encyclopedia/old_files/data/N067/N.0607770 Normed algebra
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
 +
{{TEX|auto}}
 +
{{TEX|done}}
  
 +
An [[Algebra|algebra]] over the field of real or complex numbers that is at the same time a [[Normed space|normed space]] in which multiplication satisfies some continuity condition. The simplest such condition is separate continuity. Generally speaking, this is weaker than joint continuity in the factors. For example, if one defines the algebraic operations on the set of all finite sequences  $  x = ( \epsilon _ {1} \dots \epsilon _ {n} , 0 ,\dots) $
 +
coordinate-wise and the norm by  $  \| x \| = \sum _ {k = 1 }  ^  \infty  | \epsilon _ {k} | k  ^ {-} 2 $,
 +
then there arises an algebra in which the multiplication is separately, but not jointly, continuous. The joint continuity of the multiplication in a normed algebra is equivalent to the existence of a constant  $  C $
 +
such that  $  \| xy \| \leq  C  \| x \|  \| y \| $.
 +
In this and only in this case the completion has the structure of a normed algebra that extends the original one and is a [[Banach algebra|Banach algebra]].
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1964)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.A. Naimark,  "Normed rings" , Reidel  (1964)  (Translated from Russian)</TD></TR></table>

Revision as of 08:03, 6 June 2020


An algebra over the field of real or complex numbers that is at the same time a normed space in which multiplication satisfies some continuity condition. The simplest such condition is separate continuity. Generally speaking, this is weaker than joint continuity in the factors. For example, if one defines the algebraic operations on the set of all finite sequences $ x = ( \epsilon _ {1} \dots \epsilon _ {n} , 0 ,\dots) $ coordinate-wise and the norm by $ \| x \| = \sum _ {k = 1 } ^ \infty | \epsilon _ {k} | k ^ {-} 2 $, then there arises an algebra in which the multiplication is separately, but not jointly, continuous. The joint continuity of the multiplication in a normed algebra is equivalent to the existence of a constant $ C $ such that $ \| xy \| \leq C \| x \| \| y \| $. In this and only in this case the completion has the structure of a normed algebra that extends the original one and is a Banach algebra.

Comments

References

[a1] M.A. Naimark, "Normed rings" , Reidel (1964) (Translated from Russian)
How to Cite This Entry:
Normed algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Normed_algebra&oldid=48022
This article was adapted from an original article by E.A. Gorin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article