View source for Bishop theorem

One of the early generalizations of the famous [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]], which itself is a generalization of the celebrated [[Weierstrass theorem|Weierstrass theorem]] stating that every real-valued continuous function on a closed and bounded interval is a uniform limit (cf. also [[Uniform convergence|Uniform convergence]]) of a sequence of polynomials.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301601.png" /> be a compact [[Hausdorff space|Hausdorff space]], and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301602.png" /> denote the set of all complex-valued continuous functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301603.png" /> equipped with the supremum norm, given by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301604.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301605.png" /> be a non-empty subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301606.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301607.png" /> be a non-empty subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301608.png" />. One says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b1301609.png" /> is a partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016011.png" />-anti-symmetric set if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016013.png" /> (the restriction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016014.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016015.png" />) real imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016016.png" /> is a constant. A partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016017.png" />-anti-symmetric set is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016019.png" />-anti-symmetric if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016021.png" /> purely imaginary (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016022.png" />) imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016023.png" /> is a constant.

It is easy to see that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016024.png" /> is closed under multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016025.png" /> (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016026.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016027.png" />), then every partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016028.png" />-anti-symmetric set is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016029.png" />-anti-symmetric. This is not true for arbitrary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016030.png" />. (See [[#References|[a5]]] for an example.) Every partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016031.png" />-anti-symmetric set is contained in a maximal partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016032.png" />-anti-symmetric set. Every maximal partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016033.png" />-anti-symmetric set is closed. Distinct maximal partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016034.png" />-anti-symmetric sets are disjoint. Each singleton set is a partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016035.png" />-anti-symmetric set. Thus, the family of all maximal partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016036.png" />-anti-symmetric sets forms a partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016037.png" />. Proofs of these and many other interesting properties of partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016038.png" />-anti-symmetric sets can be found in [[#References|[a5]]]. All these statements are also true for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016039.png" />-anti-symmetric sets. (See [[#References|[a5]]], [[#References|[a3]]].)

==Bishop's theorem.==


1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016040.png" /> be a uniformly closed real subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016041.png" /> containing the constant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016042.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016043.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016044.png" /> for every maximal partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016045.png" />-anti-symmetric set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016046.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016047.png" />.

2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016048.png" /> be a [[Homeomorphism|homeomorphism]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016049.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016050.png" /> is the identity mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016051.png" />. Let

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016052.png" /></td> </tr></table>

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016053.png" /> be a uniformly closed real subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016054.png" /> containing the constant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016055.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016056.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016057.png" /> for every maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016058.png" />-anti-symmetric set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016059.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016060.png" />.

In fact, 1) means that if a [[Continuous function|continuous function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016061.png" /> coincides with some function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016062.png" /> for every maximal partially <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016063.png" />-anti-symmetric set, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016064.png" />. In view of the comments preceding the statement of the theorem, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016065.png" /> is a complex subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016066.png" />, then in 1) above one can replace  "partially A-anti-symmetric"  by  "A-anti-symmetric" . This was the classical statement by E. Bishop in 1961 [[#References|[a2]]].

<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016067.png" /> is said to separate the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016068.png" />, if for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016069.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016070.png" />, there is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016071.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016072.png" />. A uniformly closed complex subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016073.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016074.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016075.png" /> and separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016076.png" /> is called a complex function algebra. Similarly, a uniformly closed real subalgebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016077.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016078.png" /> that contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016079.png" /> and separates the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016080.png" /> is called a real function algebra. In view of this, 2) is called an analogue of Bishop's theorem for real function algebras. This was proved in [[#References|[a6]]]. (See also [[#References|[a5]]].) If a complex function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016081.png" /> is closed under conjugation (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016082.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016083.png" />), then every maximal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016084.png" />-anti-symmetric set reduces to a singleton. (See [[#References|[a5]]] for a proof.) Thus, the hypotheses of Bishop's theorem are trivially satisfied by every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016085.png" />. Hence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016086.png" />. This is the classical [[Stone–Weierstrass theorem|Stone–Weierstrass theorem]], which has permeated most of modern analysis and has many generalizations. Bishop's theorem is an essential tool in proving many of these generalizations. (See [[#References|[a5]]], [[#References|[a3]]].) Similarly, if a real function algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016087.png" /> is closed under conjugation, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016088.png" />. This is an analogue of the Stone–Weierstrass theorem for real function algebras. (See [[#References|[a5]]] for a proof.)

The proof of Bishop's theorem in [[#References|[a2]]] uses many non-trivial tools from functional analysis, such as the [[Hahn–Banach theorem|Hahn–Banach theorem]], the Krein–Mil'man theorem (cf. also [[Locally convex space|Locally convex space]]) and the Riesz representation theorem (cf. [[Riesz theorem(2)|Riesz theorem]]). This proof can also be found in [[#References|[a3]]] and [[#References|[a10]]]. J.B. Prolla extended this technique to the case of vector-valued functions [[#References|[a8]]]. S. Machado formulated a quantitative version of Prolla's theorem and gave an elementary proof of it [[#References|[a7]]]. A self-contained exposition of Machado's proof can be found in [[#References|[a4]]]. In 1984, T.J. Ransford gave a very short, simple and elementary proof of Machado's version of Bishop's theorem [[#References|[a9]]] (see also [[#References|[a5]]]). This proof uses a technique also used in [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  B. Brosowski,   F. Deutsch,   "An elementary proof of the Stone–Weierstrass theorem"  ''Proc. Amer. Math. Soc.'' , '''81'''  (1981)  pp. 89–92</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E. Bishop,   "A generalization of the Stone–Weierstrass theorem"  ''Pacific J. Math.'' , '''11'''  (1961)  pp. 777–783</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R.B. Burckel,   "Characterizations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b130/b130160/b13016089.png" /> among its subalgebras" , M. Dekker  (1972)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  R.B. Burckel,   "Bishop's Stone–Weierstrass theorem"  ''Amer. Math. Monthly'' , '''91'''  (1984)  pp. 22–32</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S.H. Kulkarni,   B.V. Limaye,   "Real function algebras" , M. Dekker  (1992)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  S.H. Kulkarni,   N. Srinivasan,   "An analogue of Bishop's theorem for real function algebras"  ''Indian J. Pure Appl. Math.'' , '''18'''  (1987)  pp. 136–145</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  S. Machado,   "On Bishop's generalization of the Stone–Weierstrass theorem"  ''Indag. Math.'' , '''39'''  (1977)  pp. 218–224  ''Nederl. Akad. Wetensch. Proc. Ser. A'' , '''80'''  (1977)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  J.B. Prolla,   "Bishop's generalized Stone–Weierstrass theorem for weighted spaces"  ''Math. Ann.'' , '''191'''  (1971)  pp. 283–289</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  T.J. Ransford,   "A short elementary proof of the Bishop–Stone–Weierstrass theorem"  ''Math. Proc. Cambridge Philos. Soc.'' , '''96'''  (1984)  pp. 309–311</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  W. Rudin,   "Real and complex analysis" , McGraw-Hill  (1966)</TD></TR></table>