Stone–Weierstrass theorem#Stone–Weierstrass theorem, complex version

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) − p(x){{!}} < ε}}, or equivalently, the supremum norm {{math|{{norm|f − p}} < ε}}.}}

A constructive proof of this theorem using Bernstein polynomials is outlined on that page.

For differentiable functions, Jackson's inequality bounds the error of approximations by polynomials of a given degree: if $f$ has a continuous k-th derivative, then for every $n\backslash in\backslash mathbb\; N$ there exists a polynomial $p\_n$ of degree at most $n$ such that $\backslash lVert\; f-p\_n\backslash rVert\; \backslash leq\; \backslash frac\backslash pi\; 2\backslash frac\; 1\{(n+1)^k\}\; \backslash lVert\; f^\{(k)\}\backslash rVert$.{{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}}

However, if $f$ is merely continuous, the convergence of the approximations can be arbitrarily slow in the following sense: for any sequence of positive real numbers $(a\_n)\_\{n\backslash in\backslash mathbb\; N\}$ decreasing to 0 there exists a function $f$ such that $\backslash lVert\; f-p\backslash rVert\; >\; a\_n$ for every polynomial $p$ of degree at most $n$.{{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 }}

As a consequence of the Weierstrass approximation theorem, one can show that the space {{math|C[a, b]}} is separable: the polynomial functions are dense, and each polynomial function can be uniformly approximated by one with rational coefficients; there are only countably many polynomials with rational coefficients. Since {{math|C[a, b]}} is metrizable and separable it follows that {{math|C[a, b]}} has cardinality at most {{math|2^ℵ₀}}. (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.)

The set {{math|C[a, b]}} of continuous real-valued functions on {{math|[a, b]}}, together with the supremum norm {{math|{{norm|f}} {{=}} sup_{a ≤ x ≤ b} {{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 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 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 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.

A version of the Stone–Weierstrass theorem is also true when {{mvar|X}} is only locally compact. Let {{math|C₀(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|C₀(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 \ K}}. Again, {{math|C₀(X, R)}} is a Banach algebra with the supremum norm. A subalgebra {{mvar|A}} of {{math|C₀(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|C₀(X, R)}}. Then {{mvar|A}} is dense in {{math|C₀(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|C₀(X, R) {{=}} C(X, R)}}. There are also more general versions of the Stone–Weierstrass theorem that weaken the assumption of local compactness.{{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 }}

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| f₁, ...,  f_n }} on {{mvar|X}} and continuous functions {{math|g₁, ..., g_n}} on {{mvar|Y}} such that {{math|{{norm|f − Σ f_i g_i}} < ε}}. {{Citation needed|date=July 2018}}

Slightly more general is the following theorem, where we consider the algebra $C(X,\; \backslash Complex)$ of complex-valued continuous functions on the compact space $X$, 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 $X$ be a compact Hausdorff space and let $S$ be a separating subset of $C(X,\; \backslash Complex)$. Then the complex unital *-algebra generated by $S$ is dense in $C(X,\; \backslash Complex)$.}}

The complex unital *-algebra generated by $S$ consists of all those functions that can be obtained from the elements of $S$ 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, $f\_n\backslash to\; f$, then the real parts of those functions uniformly approximate the real part of that function, $\backslash operatorname\{Re\}f\_n\backslash to\backslash operatorname\{Re\}f$, and because for real subsets, $S\backslash subset\; C(X,\backslash Reals)\backslash subset\; C(X,\backslash Complex),$ 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_n(x) {{=}} e^2πinx, 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_n}} are an orthonormal basis of the space Lp space of square-integrable functions on {{math|[0, 1]}}. {{Citation needed|date=May 2024}}

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 $q\; =\; a\; +\; ib\; +\; jc\; +\; kd$

• its scalar part {{math|a}} is the real number $\backslash frac\{q\; -\; iqi\; -\; jqj\; -\; kqk\}\{4\}$.

Likewise

• the scalar part of {{math|−qi}} is {{math|b}} which is the real number $\backslash frac\{-qi\; -\; iq\; +\; jqk\; -\; kqj\}\{4\}$.
• the scalar part of {{math|−qj}} is {{math|c}} which is the real number $\backslash frac\{-qj\; -\; iqk\; -\; jq\; +\; kqi\}\{4\}$.
• the scalar part of {{math|−qk}} is {{math|d}} which is the real number $\backslash frac\{-qk\; +\; iqj\; -\; jqk\; -\; kq\}\{4\}$.

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 separates points.}}

The space of complex-valued continuous functions on a compact Hausdorff space $X$ i.e. $C(X,\; \backslash Complex)$ is the canonical example of a unital commutative C*-algebra $\backslash mathfrak\{A\}$. The space X may be viewed as the space of pure states on $\backslash mathfrak\{A\}$, 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 $\backslash mathfrak\{A\}$ has a C*-subalgebra $\backslash mathfrak\{B\}$ which separates the pure states of $\backslash mathfrak\{A\}$, then $\backslash mathfrak\{A\}\; =\; \backslash mathfrak\{B\}$.}}

In 1960, Jim Glimm proved a weaker version of the above conjecture.

{{math theorem | name = Stone–Weierstrass theorem (C*-algebras){{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}} | math_statement = If a unital C*-algebra $\backslash mathfrak\{A\}$ has a C*-subalgebra $\backslash mathfrak\{B\}$ which separates the pure state space (i.e. the weak-* closure of the pure states) of $\backslash mathfrak\{A\}$, then $\backslash mathfrak\{A\}=\; \backslash mathfrak\{B\}$.}}

Let {{mvar|X}} be a compact Hausdorff space. Stone's original proof of the theorem used the idea of lattices in {{math|C(X, R)}}. A subset {{mvar|L}} of {{math|C(X, R)}} is called a 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:{{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}}

{{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

1. {{mvar|B}} separates points.
2. {{mvar|B}} contains the constant function 1.
3. If {{math| f  ∈ B}} then {{math|af  ∈ B}} for all constants {{math|a ∈ R}}.
4. 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 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}}.

Another generalization of the Stone–Weierstrass theorem is due to Errett Bishop. Bishop's theorem is as follows:{{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}}

{{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_S = {g{{!}}_S : g ∈ A}}}. Suppose that {{math|f  ∈ C(X, C)}} has the following property:

{{block indent | em = 1.5 | text = {{math| f {{!}}_S ∈ A_S}} for every maximal set {{math|S ⊂ X}} such that all real functions of {{math|A_S}} 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 gives an analog for Stone–Weierstrass theorem for algebras of complex valued smooth functions on a smooth manifold.{{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}} Nachbin's theorem is as follows:{{citation|first=José G.|last=Llavona| title=Approximation of continuously differentiable functions| date=1986| publisher=North-Holland| location=Amsterdam| isbn=9780080872414}}

{{math theorem | name = Nachbin's theorem | math_statement = Let {{mvar|A}} be a subalgebra of the algebra {{math|C^∞(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 df(x)(v) ≠ 0. Then {{mvar|A}} is dense in {{math|C^∞(M)}}.}}

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.{{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}}{{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}}{{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}} According to the mathematician Yamilet Quintana, Weierstrass "suspected that any analytic functions could be represented by power series".{{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 Mathematicae18 (1995), 91–130.{{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 be represented by power series|access-date=July 3, 2021}} (arXiv 0611038v2).

• 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_n}} 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.

{{Reflist}}

• {{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}}.

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.

• {{springer|title=Stone–Weierstrass theorem|id=p/s090370}}

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