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Normed Banach Module

Let A be a normed (Banach) algebra. An algebraic left A-module X is said to be a normed (Banach) left A-module if X is a normed (Banach) space and the outer multiplication is jointly continuous, i.e., if there is a nonnegative number M such that ||ax||<=M||a||||x||,a in A,x in X. If A has an identity e, then X is called unital if ex=x for all x in X. A normed (Banach) right module can be similarly defined.

For example, every closed left ideal I of a normed algebra A can be regarded as a Banach left A-module with the product of A giving the module multiplication.

See also

Normed Banach Bimodule

This entry contributed by Mohammad Sal 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." In Handbook of Algebra, Vol. 2. Amsterdam, Netherlands: Elsevier, 1997.

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Normed Banach Module

Cite this as:

Moslehian, Mohammad Sal. "Normed Banach Module." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/NormedBanachModule.html

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