TOPICS
Search

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, Left Approximate Identity, Local Banach Algebra, Net, Normed Space, Real Normed Algebra, Right Approximate Identity

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha

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

Subject classifications