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Normed Banach Module
Let
be a normed (Banach) algebra. An algebraic left
-module
is said to be a normed (Banach) left
-module if
is a normed (Banach) space and the outer multiplication is jointly continuous, i.e., if there is a nonnegative number
such that
. If
has an identity
, then
is calledunital if
for all
. A normed (Banach) right module can be similarly defined.
For example, every closed left ideal
of a normed algebra
can be regarded as a Banach left
-module with the product of
giving the module multiplication.
See also
Normed Banach BimoduleThis entry contributed byMohammadSal Moslehian
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References
Helemskii, A. Ya.The Homology of Banach and Topological Algebras. Dordrecht, Netherlands: Kluwer, 1989.Helemskii, A. Ya. "The Homology in Algebra of Analysis." InHandbook of Algebra, Vol. 2. Amsterdam, Netherlands: Elsevier, 1997.Referenced on Wolfram|Alpha
Normed Banach ModuleCite this as:
Moslehian, Mohammad Sal. "Normed Banach Module." FromMathWorld--A Wolfram Resource, created byEric W. Weisstein.https://mathworld.wolfram.com/NormedBanachModule.html