| Stone-weierstrass Theorem |
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| continuous mappings | |
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| mathematical analysis | |
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Hausdorff Space ''K'' is considered, and instead of the Algebra of polynomial functions, approximation with elements from more general subalgebras of C(''K'') is investigated. 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 Uniformly On Compact Sets by algebras of the type appearing in the Stone–Weierstrass theorem and described below. WEIERSTRASS APPROXIMATION THEOREM The statement of the approximation theorem as originally discovered by Weierstrass is as follows: | ||
| The Set C | "''a'',''b'']" class="copylinks" target="_blank">of continuous real-valued functions on [''a'',''b'' , together with the supremum norm ''f'' = sup<sub>''x'' &isin ''f''(''x''), is a Banach Algebra , (ie an Associative Algebra and a Banach Space such that ''fg'' &le ''f'' ''g'' for all ''f'', ''g'') The set of all polynomial functions forms a subalgebra of C[''a'',''b'' , and the content of the Weierstrass approximation theorem is that this subalgebra is Dense in C[''a'',''b''] |
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| If ''f'' Is A Continuous Real-valued Function Defined On The Set | "''a'',''b'']" class="copylinks" target="_blank">x [''c'',''d'' and &epsilon>0, then there exists a polynomial function ''p'' in two variables such that ''f''(''x'',''y'') - ''p''(''x'',''y'') < &epsilon for all ''x'' in and ''y'' in [''c'',''d'' |
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| 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 | "http://wwwinformationdelightinfo/encyclopedia/entry/absolute_value" class="copylinks">Absolute Value ''f'' which in turn can be approximated by polynomials in ''f'' |
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| :Suppose ''K'' Is A Compact Hausdorff Space With At Least Two Points And ''L'' Is A Lattice In C(''K'','''R''') The Function &phi In C(''K'','''R''') Belongs To The | "http://wwwinformationdelightinfo/encyclopedia/entry/closure_(topology)" class="copylinks">Closure of ''L'' Iff for each pair of distinct points ''x'' and ''y'' in ''K'' and for each &epsilon > 0 there exists some ''f'' in ''L'' for which ''f''(''x'') - &phi(''x'') < &epsilon and ''f''(''y'') - &phi(''y'') < &epsilon |
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