Stone-Weierstrass Theorem - Revision history

Lila: /* Stone–Weierstrass theorem, real version */

2021-11-08T20:16:41Z

Stone–Weierstrass theorem, real version

Lila: /* Applications */

2021-11-08T20:13:31Z

Applications

Lila: /* Applications */

2021-11-08T20:11:44Z

Applications

Lila: /* Applications */

2021-11-08T20:09:43Z

Applications

Lila: /* Stone–Weierstrass theorem, real version */

2021-11-08T20:07:05Z

Stone–Weierstrass theorem, real version

Lila: /* Stone–Weierstrass theorem, real version */

2021-11-08T20:05:32Z

Stone–Weierstrass theorem, real version

Lila: /* Stone–Weierstrass theorem, real version */

2021-11-08T20:05:04Z

Stone–Weierstrass theorem, real version

Lila: /* Stone–Weierstrass theorem, real version */

2021-11-08T20:04:50Z

Stone–Weierstrass theorem, real version

Lila: /* Weierstrass approximation theorem */

2021-11-08T20:02:16Z

Weierstrass approximation theorem

Lila: /* Weierstrass approximation theorem */

2021-11-08T20:01:56Z

Weierstrass approximation theorem

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 :'''Stone–Weierstrass Theorem (locally compact spaces).''' 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 that weaken the assumption of local compactness.
+This version clearly implies the previous version in the case when {{mvar|X}} is compact, since in that case <span class="math-inline"><math> C_{0}(X,\R) = C(X,\R) </math></span>. There are also more general versions of the Stone–Weierstrass that weaken the assumption of local compactness.
 === Applications ===

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 The theorem has many other applications to analysis, including:
-* 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 {{math|L<sup>2</sup>([0, 1])}} of square-integrable functions on {{math|[0, 1]}}.
+* Fourier series: The set of linear combinations of functions <span class="math-inline"><math> e^{n}x = e^{2\pi inx}, n\in\Z </math></span> 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 {{math|L<sup>2</sup>([0, 1])}} of square-integrable functions on {{math|[0, 1]}}.
 == Licensing ==
 Content obtained and/or adapted from:
 * [https://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem Stone-Weierstrass theorem, Wikipedia] under a CC BY-SA license

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 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'']}}.
+* 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 <span class="math-inline"><math> |f(x,y) - p(x,y)| < \varepsilon </math></span> for all {{mvar|x}} in {{math|[''a'', ''b'']}} and {{mvar|y}} in {{math|[''c'', ''d'']}}.
 * 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 <span class="math-inline"><math> \| f - \Sigma f_i g_i \| < \varepsilon </math></span>.

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 * 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'']}}.
-* 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|{{!!}} ''f'' − Σ&nbsp;''f<sub>i</sub> g<sub>i</sub>'' {{!!}} < ''ε''}}.
+* 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 <span class="math-inline"><math> \| f - \Sigma f_i g_i \| < \varepsilon </math></span>.
 The theorem has many other applications to analysis, including:

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 == 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 <span class="math-inline"><math>  \| f\| = \text{sup}_{a\leq x\leq b} |f(x)|</math></span>, is a Banach algebra, (that is, an associative algebra and a Banach space such that {{math|{{!!}} ''fg''{{!!}} ≤ {{!!}} ''f'' {{!!}}·{{!!}}''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'']}}.
+The set {{math|C[''a'', ''b'']}} of continuous real-valued functions on {{math|[''a'', ''b'']}}, together with the supremum norm <span class="math-inline"><math>  \| f\| = \text{sup}_{a\leq x\leq b} |f(x)|</math></span>, is a Banach algebra, (that is, an associative algebra and a Banach space such that <span class="math-inline"><math> \| fg \| \leq \| f \|\cdot\| g \|</math></span> 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:

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 The statement of the approximation theorem as originally discovered by Weierstrass is as follows:
-:'''Weierstrass Approximation Theorem.''' 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 <span class="math-inline"><math>\|f(x) - p(x)\| < \varepsilon </math></span>, or equivalently, the supremum norm <span class="math-inline"><math>\|\|f - p\|\| < \varepsilon </math></span> .
+:'''Weierstrass Approximation Theorem.''' 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 <span class="math-inline"><math>|f(x) - p(x)| < \varepsilon </math></span>, or equivalently, the supremum norm <span class="math-inline"><math>\|f - p\| < \varepsilon </math></span> .
 A constructive proof of this theorem using Bernstein polynomials is outlined on that page.