Babuška–Lax–Milgram theorem: Difference between revisions
en>Francois-Pier |
|||
| Line 1: | Line 1: | ||
In [[functional analysis]], a branch of mathematics, the '''Shilov boundary''' is the smallest [[closed set|closed]] subset of the [[structure space]] of a [[commutative]] [[Banach algebra]] where an analog of the [[maximum modulus principle]] holds. It is named after its discoverer, [[Georgii Evgen'evich Shilov]]. | |||
== Precise definition and existence == | |||
Let <math>\mathcal A</math> be a [[commutative]] [[Banach algebra]] and let <math>\Delta \mathcal A</math> be its [[structure space]] equipped with the [[relative topology|relative]] [[weak topology|weak*-topology]] of the [[continuous dual space|dual]] <math>{\mathcal A}^*</math>. A closed (in this topology) subset <math>F</math> of <math>\Delta {\mathcal A}</math> is called a '''boundary''' of <math>{\mathcal A}</math> if <math>\max_{f \in \Delta {\mathcal A}} |x(f)|=\max_{f \in F} |x(f)|</math> for all <math>x \in \mathcal A</math>. | |||
The set <math>S=\bigcap\{F:F \text{ is a boundary of } {\mathcal A}\}</math> is called the '''Shilov boundary'''. It has been proved by Shilov<ref>Theorem 4.15.4 in [[Einar Hille]], [[Ralph S. Phillips]]: [http://www.ams.org/online_bks/coll31/coll31-chIV.pdf Functional analysis and semigroups]. -- AMS, Providence 1957.</ref> that <math>S</math> is a boundary of <math>{\mathcal A}</math>. | |||
Thus one may also say that Shilov boundary is the unique set <math>S \subset \Delta \mathcal A</math> which satisfies | |||
#<math>S</math> is a boundary of <math>\mathcal A</math>, and | |||
#whenever <math>F</math> is a boundary of <math>\mathcal A</math>, then <math>S \subset F</math>. | |||
== Examples == | |||
*Let <math>\mathbb D=\{z \in \mathbb C:|z|<1\}</math> be the [[open unit disc]] in the [[complex plane]] and let | |||
<math>{\mathcal A}={\mathcal H}(\mathbb D)\cap {\mathcal C}(\bar{\mathbb D})</math> be the [[disc algebra]], i.e. the functions [[holomorphic]] in <math>\mathbb D</math> and [[continuous function|continuous]] in the [[closure (topology)|closure]] of <math>\mathbb D</math> with [[supremum norm]] and usual algebraic operations. Then <math>\Delta {\mathcal A}=\bar{\mathbb D}</math> and <math>S=\{|z|=1\}</math>. | |||
== References == | |||
*{{Springer|id=B/b110310|title=Bergman-Shilov boundary}} | |||
==Notes== | |||
{{Reflist}} | |||
== See also == | |||
*[[James boundary]] | |||
[[Category:Banach algebras]] | |||
Revision as of 07:04, 28 December 2013
In functional analysis, a branch of mathematics, the Shilov boundary is the smallest closed subset of the structure space of a commutative Banach algebra where an analog of the maximum modulus principle holds. It is named after its discoverer, Georgii Evgen'evich Shilov.
Precise definition and existence
Let be a commutative Banach algebra and let be its structure space equipped with the relative weak*-topology of the dual . A closed (in this topology) subset of is called a boundary of if for all . The set is called the Shilov boundary. It has been proved by Shilov[1] that is a boundary of .
Thus one may also say that Shilov boundary is the unique set which satisfies
- is a boundary of , and
- whenever is a boundary of , then .
Examples
- Let be the open unit disc in the complex plane and let
be the disc algebra, i.e. the functions holomorphic in and continuous in the closure of with supremum norm and usual algebraic operations. Then and .
References
- Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
my web-site http://himerka.com/
Notes
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
See also
- ↑ Theorem 4.15.4 in Einar Hille, Ralph S. Phillips: Functional analysis and semigroups. -- AMS, Providence 1957.