Editing Stone–Weierstrass theorem

{{Short description|Mathematical theorem in the study of analysis}}
In [[mathematical analysis]], the '''Weierstrass approximation theorem''' states that every [[continuous function]] defined on a closed [[interval (mathematics)|interval]] {{math|[''a'', ''b'']}} can be [[uniform convergence|uniformly approximated]] as closely as desired by a [[polynomial]] function. Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in [[polynomial interpolation]]. The original version of this result was established by [[Karl Weierstrass]] in [[#Historical works|1885]] using the [[Weierstrass transform]].

[[Marshall H. Stone]] considerably generalized the theorem<ref>{{citation|first=M. H.|last=Stone|year=1937|author-link=Marshall Stone|title=Applications of the Theory of Boolean Rings to General Topology|journal=Transactions of the American Mathematical Society| volume=41| issue=3| pages=375–481| doi=10.2307/1989788| jstor=1989788| doi-access=free}}</ref> and simplified the proof.<ref>{{citation|doi=10.2307/3029750|first=M. H.|last=Stone|year=1948|author-link=Marshall Stone| title=The Generalized Weierstrass Approximation Theorem|journal=Mathematics Magazine| volume=21| issue=4| pages=167–184| jstor=3029750| mr=27121 }}; '''21''' (5), 237–254.</ref>  His result is known as the '''Stone–Weierstrass theorem'''. The Stone–Weierstrass theorem generalizes the Weierstrass approximation theorem in two directions: instead of the real interval {{math|[''a'', ''b'']}}, an arbitrary [[compact space|compact]] [[Hausdorff space]] {{mvar|X}} is considered, and instead of the [[Algebra over a field|algebra]] of polynomial functions, a variety of other families of continuous functions on <math>X</math> are shown to suffice, as is [[Stone–Weierstrass theorem#Stone–Weierstrass theorem, real version|detailed below]].  The Stone–Weierstrass theorem is a vital result in the study of the algebra of [[continuous functions on a compact Hausdorff space]].

Further, there is a generalization of the Stone–Weierstrass theorem to noncompact [[Tychonoff space]]s, namely, any continuous function on a Tychonoff space is approximated [[compact-open topology|uniformly on compact sets]] by algebras of the type appearing in the Stone–Weierstrass theorem and described below.

A different generalization of Weierstrass' original theorem is [[Mergelyan's theorem]], which generalizes it to functions defined on certain subsets of the [[complex plane]].

== Weierstrass approximation theorem ==
The statement of the approximation theorem as originally discovered by Weierstrass is as follows:

{{math theorem | name = Weierstrass approximation theorem | math_statement = Suppose {{math|''f''}} is a continuous real-valued function defined on the real interval {{math|[''a'', ''b'']}}. For every {{math|''ε'' > 0}}, there exists a polynomial {{math|''p''}} such that for all {{mvar|x}} in {{math|[''a'', ''b'']}}, we have {{math|{{!}}''f''(''x'')&nbsp;− ''p''(''x''){{!}}&nbsp;< ''ε''}}, or equivalently, the [[supremum norm]] {{math|{{norm|''f''&nbsp;− ''p''}}&nbsp;< ''ε''}}.}}

A constructive proof of this theorem using [[Bernstein polynomial]]s is outlined on that page.

=== Degree of approximation ===
For differentiable functions, [[Jackson's inequality]] bounds the error of approximations by polynomials of a given degree: if <math>f</math> has a continuous k-th derivative, then for every <math>n\in\mathbb N</math> there exists a polynomial <math>p_n</math> of degree at most <math>n</math> such that <math>\lVert f-p_n\rVert \leq \frac\pi 2\frac 1{(n+1)^k} \lVert f^{(k)}\rVert</math>.<ref>{{Cite book |last=Cheney |first=Elliott W. |title=Introduction to approximation theory |date=2000 |publisher=AMS Chelsea Publ |isbn=978-0-8218-1374-4 |edition=2. ed., repr |location=Providence, RI}}</ref>

However, if <math>f</math> is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers <math>(a_n)_{n\in\mathbb N}</math> decreasing to 0 there exists a function <math>f</math> such that <math>\lVert f-p\rVert > a_n</math> for every polynomial <math>p</math> of degree at most <math>n</math>.<ref>{{Cite journal |last=de la Cerda |first=Sofia |date=2023-08-09 |title=Polynomial Approximations to Continuous Functions |url=https://www.tandfonline.com/doi/full/10.1080/00029890.2023.2206324 |journal=The American Mathematical Monthly |language=en |volume=130 |issue=7 |pages=655 |doi=10.1080/00029890.2023.2206324 |issn=0002-9890|url-access=subscription }}</ref>

=== Applications ===
As a consequence of the Weierstrass approximation theorem, one can show that the space {{math|C[''a'', ''b'']}} is [[separable space|separable]]: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with [[rational number|rational]] coefficients; there are only [[countable|countably many]] polynomials with rational coefficients. Since {{math|C[''a'', ''b'']}} is [[metrizable space|metrizable]] and separable it follows that {{math|C[''a'', ''b'']}} has [[cardinality]] at most {{math|2<sup>ℵ<sub>0</sub></sup>}}.  (Remark: This cardinality result also follows from the fact that a continuous function on the reals is uniquely determined by its restriction to the rationals.)

== Stone–Weierstrass theorem, real version ==
The set {{math|C[''a'', ''b'']}} of continuous real-valued functions on {{math|[''a'', ''b'']}}, together with the supremum norm {{math|{{norm|''f''}} {{=}} sup<sub>''a'' ≤ ''x'' ≤ ''b''</sub> {{abs|''f'' (''x'')}}}} is a [[Banach algebra]], (that is, an [[associative algebra]] and a [[Banach space]] such that {{math|{{norm|''fg''}} ≤ {{norm|''f''}}·{{norm|''g''}}}} for all {{math| ''f'', ''g''}}). The set of all polynomial functions forms a subalgebra of {{math|C[''a'', ''b'']}} (that is, a [[linear subspace|vector subspace]] of {{math|C[''a'', ''b'']}} that is closed under multiplication of functions), and the content of the Weierstrass approximation theorem is that this subalgebra is [[Topology Glossary|dense]] in {{math|C[''a'', ''b'']}}.

Stone starts with an arbitrary compact Hausdorff space {{mvar|X}} and considers the algebra {{math|C(''X'', '''R''')}} of real-valued continuous functions on {{mvar|X}}, with the topology of [[uniform convergence]]. He wants to find subalgebras of {{math|C(''X'', '''R''')}} which are dense. It turns out that the crucial property that a subalgebra must satisfy is that it ''[[separating set|separates points]]'': a set {{mvar|A}} of functions defined on {{mvar|X}} is said to separate points if, for every two different points {{mvar|x}} and {{mvar|y}} in {{mvar|X}} there exists a function {{mvar|p}} in {{mvar|A}} with {{math|''p''(''x'') ≠ ''p''(''y'')}}. Now we may state:

{{math theorem | name = Stone–Weierstrass theorem (real numbers) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space and {{mvar|A}} is a subalgebra of {{math|C(''X'', '''R''')}} which contains a non-zero constant function. Then {{mvar|A}} is dense in {{math|C(''X'', '''R''')}} [[if and only if]] it separates points.}}

This implies Weierstrass' original statement since the polynomials on {{math|[''a'', ''b'']}} form a subalgebra of {{math|C[''a'', ''b'']}} which contains the constants and separates points.

=== Locally compact version ===
A version of the Stone–Weierstrass theorem is also true when {{mvar|X}} is only [[locally compact]]. Let {{math|C<sub>0</sub>(''X'', '''R''')}} be the space of real-valued continuous functions on {{mvar|X}} that [[vanish at infinity]]; that is, a continuous function {{math| ''f'' }} is in {{math|C<sub>0</sub>(''X'', '''R''')}} if, for every {{math|''ε'' > 0}}, there exists a compact set {{math|''K'' ⊂ ''X''}} such that {{math| {{abs|''f''}}  < ''ε''}} on {{math|''X''&nbsp;\&nbsp;''K''}}. Again, {{math|C<sub>0</sub>(''X'', '''R''')}} is a [[Banach algebra]] with the [[supremum norm]]. A subalgebra {{mvar|A}} of {{math|C<sub>0</sub>(''X'', '''R''')}} is said to '''vanish nowhere''' if not all of the elements of {{mvar|A}} simultaneously vanish at a point; that is, for every {{mvar|x}} in {{mvar|X}}, there is some {{math| ''f'' }} in {{mvar|A}} such that {{math| ''f'' (''x'') ≠ 0}}. The theorem generalizes as follows:

{{math theorem | name = Stone–Weierstrass theorem (locally compact spaces) | math_statement = Suppose {{mvar|X}} is a ''locally compact'' Hausdorff space and {{mvar|A}} is a subalgebra of {{math|C<sub>0</sub>(''X'', '''R''')}}. Then {{mvar|A}} is dense in {{math|C<sub>0</sub>(''X'', '''R''')}} (given the topology of [[uniform convergence]]) if and only if it separates points and vanishes nowhere.}}

This version clearly implies the previous version in the case when {{mvar|X}} is compact, since in that case {{math|C<sub>0</sub>(''X'', '''R''') {{=}} C(''X'', '''R''')}}. There are also more general versions of the Stone–Weierstrass theorem that weaken the assumption of local compactness.<ref name=Willard>{{cite book |first=Stephen |last=Willard |title=General Topology |url=https://archive.org/details/generaltopology00will_0 |url-access=registration |page=[https://archive.org/details/generaltopology00will_0/page/293 293] |publisher=Addison-Wesley |year=1970 |isbn=0-486-43479-6 }}</ref>

=== Applications ===
The Stone–Weierstrass theorem can be used to prove the following two statements, which go beyond Weierstrass's result.
* If {{math| ''f'' }} is a continuous real-valued function defined on the set {{math|[''a'', ''b''] × [''c'', ''d'']}} and {{math|''ε'' > 0}}, then there exists a polynomial function {{mvar|p}} in two variables such that {{math|{{!}} ''f'' (''x'', ''y'') − ''p''(''x'', ''y'') {{!}} < ''ε''}} for all {{mvar|x}} in {{math|[''a'', ''b'']}} and {{mvar|y}} in {{math|[''c'', ''d'']}}.{{Citation needed|date=July 2018}}
* If {{mvar|X}} and {{mvar|Y}} are two compact Hausdorff spaces and {{math|''f'' : ''X'' × ''Y'' → '''R'''}} is a continuous function, then for every {{math|''ε'' > 0}} there exist {{math|''n'' > 0}} and continuous functions {{math| ''f''<sub>1</sub>, ...,  ''f<sub>n</sub>'' }} on {{mvar|X}} and continuous functions {{math|''g''<sub>1</sub>, ..., ''g<sub>n</sub>''}} on {{mvar|Y}} such that {{math|{{norm|''f'' − Σ&nbsp;''f<sub>i</sub> g<sub>i</sub>''}} < ''ε''}}. {{Citation needed|date=July 2018}}

== Stone–Weierstrass theorem, complex version ==
Slightly more general is the following theorem, where we consider the algebra  <math>C(X, \Complex)</math> of complex-valued continuous functions on the compact space <math>X</math>, again with the topology of uniform convergence. This is a [[C*-algebra]] with the *-operation given by pointwise [[complex conjugation]].

{{math theorem | name = Stone–Weierstrass theorem (complex numbers) | math_statement = Let <math>X</math> be a compact Hausdorff space and let <math>S</math> be a [[separating set|separating subset]] of <math>C(X, \Complex)</math>. Then the complex [[unital algebra|unital]] [[*-algebra]] generated by <math>S</math> is dense in <math>C(X, \Complex)</math>.}}

The complex unital *-algebra generated by <math>S</math> consists of all those functions that can be obtained from the elements of <math>S</math> by throwing in the constant function {{math|1}} and adding them, multiplying them, conjugating them, or multiplying them with complex scalars, and repeating finitely many times.

This theorem implies the real version, because if a net of complex-valued functions uniformly approximates a given function, <math>f_n\to f</math>, then the real parts of those functions uniformly approximate the real part of that function, <math>\operatorname{Re}f_n\to\operatorname{Re}f</math>, and because for real subsets, <math>S\subset C(X,\Reals)\subset C(X,\Complex),</math> taking the real parts of the generated complex unital (selfadjoint) algebra agrees with the generated real unital algebra generated.

As in the real case, an analog of this theorem is true for locally compact Hausdorff spaces.

The following is an application of this complex version.
* [[Fourier series]]: The set of linear combinations of functions {{math|''e<sub>n</sub>''(''x'') {{=}} ''e''<sup>2''πinx''</sup>, ''n'' ∈ '''Z'''}} is dense in {{math|C([0, 1]/{0, 1})}}, where we identify the endpoints of the interval {{math|[0, 1]}} to obtain a circle.  An important consequence of this is that the {{math|''e<sub>n</sub>''}} are an [[orthonormal basis]] of the space [[Lp space|{{math|L<sup>2</sup>([0, 1])}}]] of [[square-integrable function]]s on {{math|[0, 1]}}. {{Citation needed|date=May 2024}}

== Stone–Weierstrass theorem, quaternion version ==
Following {{harvtxt|Holladay|1957}}, consider the algebra {{math|C(''X'', '''H''')}} of quaternion-valued continuous functions on the compact space {{mvar|X}}, again with the topology of uniform convergence. 

If a quaternion {{math|''q''}} is written in the form <math display=inline>q = a + ib + jc + kd</math>
*its scalar part {{math|''a''}} is the real number <math>\frac{q - iqi - jqj - kqk}{4}</math>.
Likewise 
*the scalar part of {{math|−''qi''}} is {{math|''b''}} which is the real number <math>\frac{-qi - iq + jqk - kqj}{4}</math>.
*the scalar part of {{math|−''qj''}} is {{math|''c''}} which is the real number <math>\frac{-qj - iqk - jq + kqi}{4}</math>.
*the scalar part of {{math|−''qk''}} is {{math|''d''}} which is the real number <math>\frac{-qk + iqj - jqk - kq}{4}</math>.

Then we may state:
{{math theorem | name = Stone–Weierstrass theorem (quaternion numbers) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space and {{mvar|A}} is a subalgebra of {{math|C(''X'', '''H''')}} which contains a non-zero constant function. Then {{mvar|A}} is dense in {{math|C(''X'', '''H''')}} if and only if it [[separating set|separates points]].}}

== Stone–Weierstrass theorem, C*-algebra version ==
The space of complex-valued continuous functions on a compact Hausdorff space <math>X</math> i.e. <math>C(X, \Complex)</math> is the canonical example of a unital [[C*-algebra#Commutative C.2A-algebras|commutative C*-algebra]] <math>\mathfrak{A}</math>. The space ''X'' may be viewed as the space of pure states on <math>\mathfrak{A}</math>, with the weak-* topology. Following the above cue, a non-commutative extension of the Stone–Weierstrass theorem, which remains unsolved, is as follows:

{{math theorem | name = Conjecture | math_statement = If a unital [[C*-algebra]] <math>\mathfrak{A}</math> has a C*-subalgebra <math>\mathfrak{B}</math> which separates the pure states of <math>\mathfrak{A}</math>, then <math>\mathfrak{A} = \mathfrak{B}</math>.}}

In 1960, [[James Glimm|Jim Glimm]] proved a weaker version of the above conjecture.

{{math theorem | name = Stone–Weierstrass theorem (C*-algebras)<ref>{{cite journal |first=James |last=Glimm |author-link=James Glimm |title=A Stone–Weierstrass Theorem for C*-algebras |journal=[[Annals of Mathematics]] |series=Second Series |volume=72 |issue=2 |year=1960 |pages=216–244 [Theorem 1] |jstor=1970133 |doi=10.2307/1970133}}</ref> | math_statement = If a unital C*-algebra <math>\mathfrak{A}</math> has a C*-subalgebra <math>\mathfrak{B}</math> which separates the pure state space (i.e. the weak-* closure of the pure states) of <math>\mathfrak{A}</math>, then <math> \mathfrak{A}= \mathfrak{B}</math>.}}

== Lattice versions ==
Let {{mvar|X}} be a compact Hausdorff space. Stone's original proof of the theorem used the idea of [[lattice (order)|lattices]] in {{math|C(''X'', '''R''')}}.  A subset {{mvar|L}} of {{math|C(''X'', '''R''')}} is called a [[lattice (order)|lattice]] if for any two elements {{math| ''f'', ''g'' ∈ ''L''}}, the functions {{math|max{ ''f'', ''g''}, min{ ''f'', ''g''} }}also belong to {{mvar|L}}. The lattice version of the Stone–Weierstrass theorem states:

{{math theorem | name = Stone–Weierstrass theorem (lattices) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space with at least two points and {{mvar|L}} is a lattice in {{math|C(''X'', '''R''')}} with the property that for any two distinct elements {{mvar|x}} and {{mvar|y}} of {{mvar|X}} and any two real numbers {{mvar|a}} and {{mvar|b}} there exists an element {{math| ''f''  ∈ ''L''}} with {{math| ''f'' (''x'') {{=}} ''a''}} and {{math| ''f'' (''y'') {{=}} ''b''}}. Then {{mvar|L}} is dense in {{math|C(''X'', '''R''')}}.}}

The above versions of Stone–Weierstrass can be proven from this version once one realizes that the lattice property can also be formulated using the [[absolute value]] {{math|{{!}} ''f'' {{!}}}} which in turn can be approximated by polynomials in {{math| ''f'' }}. A variant of the theorem applies to linear subspaces of {{math|C(''X'', '''R''')}} closed under max:<ref>{{citation|first1=E|last1=Hewitt|author-link=Edwin Hewitt | first2=K|last2=Stromberg | title=Real and abstract analysis| year=1965|publisher=Springer-Verlag | at = Theorem 7.29}}</ref>

{{math theorem | name = Stone–Weierstrass theorem (max-closed) | math_statement = Suppose {{mvar|X}} is a compact Hausdorff space and {{mvar|B}} is a family of functions in {{math|C(''X'', '''R''')}} such that
# {{mvar|B}} separates points.
# {{mvar|B}} contains the constant function 1.
# If {{math| ''f''  ∈ ''B''}} then {{math|''af''  ∈ ''B''}} for all constants {{math|''a'' ∈ '''R'''}}.
# If {{math| ''f'',  ''g'' ∈ ''B''}}, then {{math| ''f''  + ''g'', max{ ''f'', ''g''} ∈ ''B''}}.
Then {{mvar|B}} is dense in {{math|C(''X'', '''R''')}}.}}

More precise information is available:

:Suppose {{mvar|X}} is a compact Hausdorff space with at least two points and {{mvar|L}} is a lattice in {{math|C(''X'', '''R''')}}. The function {{math|''φ'' ∈ C(''X'', '''R''')}} belongs to the [[closure (topology)|closure]] of {{mvar|L}} if and only if for each pair of distinct points ''x'' and ''y'' in {{mvar|X}} and for each {{math|''ε'' > 0}} there exists some {{math| ''f''  ∈ ''L''}} for which {{math|{{!}} ''f'' (''x'') − ''φ''(''x''){{!}} < ''ε''}} and {{math|{{!}} ''f'' (''y'') − ''φ''(''y''){{!}} < ''ε''}}.

== Bishop's theorem ==
Another generalization of the Stone–Weierstrass theorem is due to [[Errett Bishop]].  Bishop's theorem is as follows:<ref>{{citation|first=Errett|last=Bishop|author-link=Errett Bishop|title=A generalization of the Stone–Weierstrass theorem| journal=Pacific Journal of Mathematics|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?handle=euclid.pjm/1103037116&view=body&content-type=pdf_1| year=1961| volume=11| issue=3| pages=777–783| doi=10.2140/pjm.1961.11.777|doi-access=free}}</ref>

{{math theorem | name = Bishop's theorem | math_statement = Let {{mvar|A}} be a closed subalgebra of the complex [[Banach algebra]] {{math|C(''X'', '''C''')}} of continuous complex-valued functions on a compact Hausdorff space {{mvar|X}}, using the supremum norm. For  {{math|''S'' ⊂ ''X''}}  we write {{math|1=''A<sub>S</sub>'' = {''g{{!}}<sub>S</sub>'' : g &isin; ''A''}<nowiki/>}}. Suppose that {{math|''f''  ∈ C(''X'', '''C''')}} has the following property:
{{block indent | em = 1.5 | text = {{math| ''f'' {{!}}<sub>''S''</sub> ∈ ''A<sub>S</sub>''}} for every maximal set {{math|''S'' ⊂ ''X''}} such that all real functions of {{math|''A<sub>S</sub>''}} are constant.}}
Then {{math| ''f''  ∈ ''A''}}.}}

{{harvtxt|Glicksberg|1962}} gives a short proof of Bishop's theorem using the [[Krein–Milman theorem]] in an essential way, as well as the [[Hahn–Banach theorem]]: the process of {{harvtxt|Louis de Branges|1959}}.  See also {{harvtxt|Rudin|1973|loc=§5.7}}.

== Nachbin's theorem ==
Nachbin's theorem gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold.<ref>{{citation|first=L.|last=Nachbin|title=Sur les algèbres denses de fonctions diffèrentiables sur une variété|journal=C. R. Acad. Sci. Paris|date=1949|volume=228|pages=1549–1551}}</ref> Nachbin's theorem is as follows:<ref>{{citation|first=José G.|last=Llavona| title=Approximation of continuously differentiable functions| date=1986| publisher=North-Holland| location=Amsterdam| isbn=9780080872414}}</ref>

{{math theorem | name = Nachbin's theorem | math_statement = Let {{mvar|A}} be a subalgebra of the algebra {{math|C<sup>∞</sup>(''M'')}} of smooth functions on a finite dimensional smooth manifold {{mvar|M}}. Suppose that {{mvar|A}} separates the points of {{mvar|M}} and also separates the tangent vectors of {{mvar|M}}: for each point ''m'' ∈ ''M'' and tangent vector ''v'' at the tangent space at ''m'', there is a ''f'' ∈ {{mvar|A}} such that d''f''(''x'')(''v'') ≠ 0. Then {{mvar|A}} is dense in {{math|C<sup>∞</sup>(''M'')}}.}}

==Editorial history==
In 1885 it was also published in an English version of the paper whose title was ''On the possibility of giving an analytic representation to an arbitrary function of real variable''.<ref>{{cite journal| first1=Allan|last1=Pinkus| url=https://www.math.technion.ac.il/hat/fpapers/wap.pdf|title=Weierstrass and Approximation Theory|page=8|journal= Journal of Approximation Theory|volume=107|issue=1|oclc=4638498762|issn=0021-9045|access-date=July 3, 2021|archive-url=https://web.archive.org/web/20131019070518/https://www.math.technion.ac.il/hat/fpapers/wap.pdf|archive-date=October 19, 2013| url-status=live}}</ref><ref>{{cite journal|first1=Allan|last1=Pinkus|url=https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.62.520&rep=rep1&type=pdf|title=Density methods and results in approximation theory|oclc=200133324|issn=0137-6934|journal=Orlicz Centenary Volume|series=Banach Center publications |volume= 64|year=2004|publisher=Institute of Mathematics, [[Polish Academy of Sciences]]|page=3|citeseerx=10.1.1.62.520|archive-url=https://web.archive.org/web/20210703202612/https://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.62.520&rep=rep1&type=pdf|archive-date=July 3, 2021|url-status=live}}</ref><ref>{{cite book|first1=Zbigniew|last1=Ciesielski|author-link1=Zbigniew Ciesielski|first2=Aleksander|last2=Pełczyński|author-link2=Aleksander Pelczynski|first3= Leszek|last3= Skrzypczak| url=https://www.google.it/search?tbm=bks&hl=it&q=%22On+the+possibility+of+giving+an+analytic+representation+to+an+arbitrary%22+weierstrass |title=Orlicz centenary volume : proceedings of the conferences "The Wladyslaw Orlicz Centenary Conference" and Function Spaces VII : Poznan, 20-25 July 2003. Vol. I, Plenary lectures|page=175|series= Banach Center publications|volume=64|publisher= Institute of Mathematics. Polish Academy of Sciences|year=2004|oclc=912348549}}</ref><ref name="arxiv_0611038v2" /><ref name="arxiv_0611034v3" /> According to the mathematician [[Lorenzo Mendoza Fleury Science Prize|Yamilet Quintana]], Weierstrass "suspected that any [[analytic function|analytic functions]] could be represented by [[power series#Analytic functions|power series]]".<ref name="arxiv_0611034v3">{{cite journal|first1=Yamilet|last1= Quintana|title=On Hilbert extensions of Weierstrass' theorem with weights|page=202|journal=Journal of Function Spaces|volume=8|issue=2|year=2010|publisher=Scientific Horizon|oclc=7180746563 |issn=0972-6802|doi=10.1155/2010/645369|doi-access= free|arxiv=math/0611034}} (arXiv 0611034v3). Citing: D. S. Lubinsky, ''Weierstrass' Theorem in the twentieth century: a selection'', in ''Quaestiones Mathematicae''18 (1995), 91–130.</ref><ref name="arxiv_0611038v2">{{cite journal|first1=Yamilet|last1= Quintana|author2=Perez D.| url=https://www.academia.edu/37183861|title=A survey on the Weierstrass approximation theorem|journal=Divulgaciones Matematicas| volume=16|issue= 1|year=2008|page=232|oclc=810468303|quote=Weierstrass' perception on analytic functions was of functions that could berepresented by power series|access-date=July 3, 2021}} (arXiv 0611038v2).</ref>

== See also ==
*[[Müntz–Szász theorem]]
* [[Bernstein polynomial]]
* [[Runge's phenomenon]] shows that finding a polynomial {{mvar|P}} such that {{math| ''f'' (''x'') {{=}} ''P''(''x'')}} for some finely spaced {{math|''x'' {{=}} ''x<sub>n</sub>''}} is a bad way to attempt to find a polynomial approximating {{math| ''f'' }} uniformly. A better approach, explained e.g. in {{harvtxt|Rudin|1976}}, p. 160, eq. (51) ff., is to construct polynomials {{mvar|P}} uniformly approximating {{math| ''f'' }} by taking the convolution of {{math| ''f'' }} with a family of suitably chosen polynomial kernels.
* [[Mergelyan's theorem]], concerning polynomial approximations of complex functions.

== Notes ==
{{Reflist}}

== References ==
* {{citation | last =Holladay |first=John C. | title =The Stone–Weierstrass theorem for quaternions| journal = Proc. Amer. Math. Soc. | volume =8| year =1957|url =http://www.ams.org/journals/proc/1957-008-04/S0002-9939-1957-0087047-7/S0002-9939-1957-0087047-7.pdf| doi=10.1090/S0002-9939-1957-0087047-7| pages=656| doi-access =free}}.
* {{citation | last =Louis de Branges |author-link=Louis de Branges | title =The Stone–Weierstrass theorem| journal = Proc. Amer. Math. Soc. | volume = 10 | issue=5 | year =1959| pages =822–824 | doi=10.1090/s0002-9939-1959-0113131-7| doi-access =free }}.
* [[Jan Brinkhuis]] & Vladimir Tikhomirov (2005) ''Optimization: Insights and Applications'', [[Princeton University Press]] {{isbn|978-0-691-10287-0}} {{mr|id=2168305}}.
* {{citation|first=James|last=Glimm|title=A Stone–Weierstrass Theorem for C*-algebras|jstor=1970133|journal=Annals of Mathematics |series=Second Series|volume=72|issue=2|year=1960|pages=216–244|doi=10.2307/1970133}}
* {{citation|last=Glicksberg|first=Irving|title=Measures Orthogonal to Algebras and Sets of Antisymmetry| journal=Transactions of the American Mathematical Society| year=1962| volume=105| issue=3| pages=415–435| doi=10.2307/1993729| jstor=1993729|doi-access=free}}.
* {{citation|first=Walter|last=Rudin|author-link=Walter Rudin|title=Principles of mathematical analysis|publisher=McGraw-Hill|year=1976|isbn=978-0-07-054235-8|edition=3rd}}
* {{citation|first=Walter|last=Rudin|author-link=Walter Rudin|title=Functional analysis|publisher=McGraw-Hill| year=1973| isbn=0-07-054236-8|url-access=registration|url=https://archive.org/details/functionalanalys00rudi}}.
* {{citation | last =JG Burkill| title =Lectures On Approximation By Polynomials| url =http://www.math.tifr.res.in/~publ/ln/tifr16.pdf}}.

=== Historical works ===
The historical publication of Weierstrass (in [[German language]]) is freely available from the digital online archive of the ''[http://bibliothek.bbaw.de/ Berlin Brandenburgische Akademie der Wissenschaften]'':

* K. Weierstrass (1885). Über die analytische Darstellbarkeit sogenannter willkürlicher Functionen einer reellen Veränderlichen. ''Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin'', 1885 (II). {{pb}} [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1885-2&seite:int=109 Erste Mitteilung] (part 1) pp. 633–639, [http://bibliothek.bbaw.de/bibliothek-digital/digitalequellen/schriften/anzeige/index_html?band=10-sitz/1885-2&seite:int=272 Zweite Mitteilung] (part 2) pp. 789–805.

== External links ==
* {{springer|title=Stone–Weierstrass theorem|id=p/s090370}}
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{{DEFAULTSORT:Stone-Weierstrass Theorem}}
[[Category:Theory of continuous functions]]
[[Category:Theorems in mathematical analysis]]
[[Category:Theorems in approximation theory]]
[[Category:1885 in science]]
[[Category:1937 in science]]
[[Category:19th century in mathematics]]
[[Category:20th century in mathematics]]