Bishop's Theorem

The set of Bishop is a tenet of the mathematical branch of functional analysis , of a work of the US mathematician Errett Bishop back from the year 1961st It is closely connected to Stone-Weierstrass's approximation theorem , which it has as a direct consequence and thus generalizes. Bishop's theorem can be derived from the theorems of Krein-Milman , Hahn-Banach and Banach-Alaoglu .

Formulation of the sentence

It can be specified as follows:

A compact Hausdorff space and the function algebra of the continuous complex-valued functions are given . ${\displaystyle X\neq \emptyset }$ ${\displaystyle C=C(X,\mathbb {C} )}$ ${\displaystyle f\colon X\to \mathbb {C} }$

A closed sub-algebra is given therein and further one . ${\displaystyle A\subseteq C}$ ${\displaystyle g\in C}$

${\displaystyle A}$contain the constant functions and, in addition, the following condition applies:

If there is any maximal -antisymmetric subset , there is always one with for all . ${\displaystyle E\subseteq X}$ ${\displaystyle A}$ ${\displaystyle f\in A}$ ${\displaystyle g(x)=f(x)}$ ${\displaystyle x\in E}$

Then is . ${\displaystyle g\in A}$

Explanations and Notes

• The function algebra is as usual with the supremum provided. ${\displaystyle C}$
• Seclusion within the function algebra in the sense of from the supremum arising topology of uniform convergence to understand. ${\displaystyle C}$
• In the function algebra is exactly then a subalgebra if a linear subspace of and also has the property that for any two , and always for the by complex multiplication resulting function in included is. ${\displaystyle C}$ ${\displaystyle A\subseteq C}$ ${\displaystyle A}$ ${\displaystyle C}$ ${\displaystyle f_{1}\in A}$ ${\displaystyle f_{2}\in A}$ ${\displaystyle f_{1}f_{2}\colon X\to \mathbb {C} \;,\;x\mapsto f_{1}(x)f_{2}(x)\;,\;}$ ${\displaystyle A}$
• A subset is called -antisymmetric if each with is always a constant function. ${\displaystyle E\subseteq X}$ ${\displaystyle A}$ ${\displaystyle f\in A}$ ${\displaystyle f(E)\subseteq {\mathbb {R} }}$
• A maximum -antisymmetric subset is one that is not really included by any other -antisymmetric subset . ${\displaystyle A}$ ${\displaystyle A}$
• Every maximal -antisymmetric subset is closed within the topological space . ${\displaystyle A}$ ${\displaystyle X}$
• The set system of all maximal -antisymmetric subsets forms a decomposition of . ${\displaystyle A}$ ${\displaystyle X}$
• Stone-Weierstrass's approximation theorem is obtained from Bishop's theorem, taking into account that, because of the assumptions made in the approximation theorem, no -antisymmetric subset can contain two or more points . ${\displaystyle A}$

Machado's lemma

The Brazilian mathematician Silvio Machado provided a lemma for Bishop's theorem and for Stone-Weierstrass's approximation theorem , with which he derived and generalized these results in a new way. It arises in a non-constructive way, namely using Zorn's lemma . Machado's lemma can be stated as follows:

Let a Hausdorff space and the function algebra of the infinitely vanishing continuous functions be given , with the field of real numbers or the field of complex numbers . ${\displaystyle X\neq \emptyset }$ ${\displaystyle C_{0}=C_{0}(X,\mathbb {K} )}$ ${\displaystyle f\colon X\to \mathbb {K} }$ ${\displaystyle {\mathbb {K} }}$

Furthermore, be a closed subalgebra of and . ${\displaystyle A}$ ${\displaystyle C_{0}}$ ${\displaystyle f_{0}\in C_{0}}$

Then:

There is a non-empty, closed antisymmetric subset with the property that the equation is satisfied with regard to the associated distance functions . ${\displaystyle A}$ ${\displaystyle E\subseteq X}$ ${\displaystyle {\operatorname {dist} }_{E}(f_{0},A)={\operatorname {dist} }_{X}(f_{0},A)}$

Explanations and Notes

• In function algebra, the same conditions apply as above with regard to norm and topology. ${\displaystyle C_{0}}$
• We say of a (continuous) function that it vanishes at infinity if for any positive number there is a compact subset such that for is always satisfied. ${\displaystyle f\colon X\to \mathbb {K} }$ ${\displaystyle \epsilon }$ ${\displaystyle C\subseteq X}$ ${\displaystyle x\in {X\setminus C}}$ ${\displaystyle |f(x)|<\epsilon }$
• For a subset and a function is here , which means and the absolute value function is. ${\displaystyle S\subseteq X}$ ${\displaystyle f\in C_{0}}$ ${\displaystyle {\operatorname {dist} }_{S}(f,A)=\inf _{g\in A}\|f-g\|_{S}}$ ${\displaystyle \|f\|_{S}=\sup _{x\in S}|f(x)|}$ ${\displaystyle |\cdot |}$

A generalized version of Stone-Weierstrass's approximation theorem

It says:

If the closed subalgebra appearing in Machado's lemma has the general properties mentioned in the approximation theorem , then is . ${\displaystyle A\subseteq C_{0}}$ ${\displaystyle A=C_{0}}$

This means:.

For any closed subalgebra that has the following three properties, namely: ${\displaystyle A\subseteq C_{0}}$

1. that for every two different one exists with , ${\displaystyle x,y\in X}$ ${\displaystyle f\in A}$ ${\displaystyle f(x)\neq f(y)}$

2. that at any one exists with , ${\displaystyle x\in X}$ ${\displaystyle g\in A}$ ${\displaystyle g(x)\neq 0}$

3. that - in the event - with each and the associated complex conjugate function in included is ${\displaystyle \mathbb {K} =\mathbb {C} }$ ${\displaystyle f\in A}$ ${\displaystyle {\overline {f}}\colon X\to \mathbb {K} ,x\mapsto {\overline {f(x)}},}$ ${\displaystyle A}$

also applies . ${\displaystyle A=C_{0}}$

literature

• Errett Bishop: A generalization of the Stone-Weierstrass theorem . In: Pacific Journal of Mathematics . tape 11 , 1961, pp. 777-783 ( MR0133676 ).
• Silvio Machado: On Bishop's generalization of the Weierstrass-Stone theorem . In: Indagationes Mathematicae . tape 39 , 1977, pp. 218-224 ( MR0448046 ).
• Friedrich Hirzebruch , Winfried Scharlau : Introduction to Functional Analysis (=  series "BI University Pocket Books" . Volume 296 ). Bibliographisches Institut , Mannheim, Vienna, Zurich 1971, ISBN 3-411-00296-4 ( MR0463864 ).
• Thomas J. Ransford : A short elementary proof of the Bishop-Stone-Weierstrass theorem . In: Mathematical Proceedings of the Cambridge Philosophical Society . tape 96 , 1984, pp. 309-311 ( MR0757664 ).
• Walter Rudin : Functional Analysis (=  International Series in Pure and Applied Mathematics ). 2nd Edition. McGraw-Hill , Boston (et al.) 1991, ISBN 0-07-054236-8 ( MR1157815 ).
• Mícheál Ó Searcóid : Elements of Abstract Analysis (=  Springer Undergraduate Mathematics Series . Volume 15 ). Springer Verlag , London ( inter alia ) 2002, ISBN 1-85233-424-X ( MR1870768 ).
• Stephen Willard : General Topology (=  Addison-Wesley Series in Mathematics ). Addison-Wesley , Reading, Massachusetts (et al.) 1970 ( MR0264581 ).

Individual evidence

1. ^ Walter Rudin: Functional Analysis. 1991, p. 121 ff
2. ↑ Rudin, op.cit., P. 121
3. ↑ Mícheál Ó Searcóid: Elements of Abstract Analysis. 2002, p. 241
4. ↑ Ó Searcóid, op.cit., P. 243