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Approximate Identity

If A is a normed algebra, a net {e_i} in A is called an approximate identity for A if

sup_(i)|e_i|<infty

and if for each a in A, e_ia->a and ae_i->a. Though this definition makes sense framed with respect to any normed algebra A, it is usually delivered in the specific cases that A is either a Banach algebra or a local Banach algebra.

Note that if only e_ia->a (ae_i->a, respectively), then {e_i} is said to be a left (respectively, right) approximate identity.

See also

Algebra, Banach Algebra, Local Banach Algebra, Net, Normed Space, Real Normed Algebra

This entry contributed by Christopher Stover

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References

Blackadar, B. K-Theory for Operator Algebras. New York: Cambridge University Press, 1998.Conway, J. A Course in Functional Analysis. New York: Springer-Verlag, 1990.

Cite this as:

Stover, Christopher. "Approximate Identity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/ApproximateIdentity.html

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