Editing Stone–Weierstrass theorem (section)

== 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.)