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Banach Completion

For a normed space (X,||·||), define X^~ to be the set of all equivalent classes of Cauchy sequences obtained by the relation

{x_n}∼{y_n} if and only if lim_(n)||x_n-y_n||=0.
(1)

For x^~=[{x_n}] and y^~=[{y_n}], let

x^~+y^~=[{x_n+y_n}]
(2)
lambdax^~=[{lambdax_n}]
(3)
||x^~||_∼=lim_(n)||x_n||.
(4)

Then (X^~,||·||_∼) is a Banach space containing a dense subspace that is isometric with X. X^~ is called the (Banach) completion of X (Kreyszig 1978).

If A is a normed algebra, a^~b^~=[{a_nb_n}] makes A^~ into a Banach algebra. Moreover, if A is a pre-C^*-algebra then A^~ equipped with (a^~)^*=lim_(n)a_n^* is a C^*-algebra (Murphy 1990).

This entry contributed by Mohammad Sal Moslehian

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References

Kreyszig, E. Introductory Functional Analysis with Applications. New York: Wiley, 1978.Murphy, G. J. C-*-Algebras and Operator Theory. New York: Academic Press, 1990.

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Banach Completion

Cite this as:

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

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