Banach space

From Citizendium
Revision as of 03:25, 18 December 2007 by imported>Hendra I. Nurdin (New)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search
This article is developing and not approved.
Main Article
Discussion
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and subject to a disclaimer.

In mathematics, particularly in the branch known as functional analysis, a Banach space is a complete normed space. It is named after famed Hungarian-Polish mathematician Stefan Banach.

The space of all continous complex (resp. real) linear functionals of a complex (resp. real) Banach space is called its dual space. This dual space is also a Banach space under when endowed with the operator norm on the continuous (hence, bounded) linear functionals.

Examples of Banach spaces

1. The Euclidean space with any norm is a Banach space. More generally, any finite dimensional normed space is a Banach space (due to its isomorphism to some Euclidean space).

2. Let , , denote the space of all complex-valued measurable function on the unit circle of the complex plane (with respect to the Haar measure on ) satisfying:

,

if , or

if . Then is a Banach space with a norm defined by

,

if , or

if . The case p = 2 is special since it is also a Hilbert space and is in fact the only Hilbert space among the spaces, .

Further reading

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980