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Mathematical Analysis

Sam Macdonald

Section 7.6 The Stone-Weierstrass Theorem

Basic problem: Approximate a continuous function \(f:X\rightarrow\mathbb{C}\) uniformly by a “nice” sequence of functions \(f_n:X\rightarrow\mathbb{C}\text{.}\)

Convention 7.28.

Let \(X\) be a compact metric space. We'll write \(C(X)\) for either \(C(X,\mathbb{C})\) or \(C(X,\mathbb{R})\) and view \(C(X)\) as either a complex or real vector space, respectively.

Definition 7.29.

A vector subspace \(A\subseteq C(X)\) is called
  • A subalgebra if \(f,g\in A\implies fg\in A\)
  • unital if \(1\in A\) (1=constant function \(x\mapsto 1\))
  • self-adjoint if \(f\in A\implies\overline{f}\in A\text{.}\)

Remark 7.30.

In the real case, every subspace of \(C(X,\mathbb{R})\) is self-adjoint.

Example 7.31.

  • \(A=C(X)\) is a subalebra of \(C(X)\) and is unital, self-adjoint
  • \(\{{\text{constant functions}}\}\subseteq C(X)\) is a unital self-adjoint subalgebra.
  • \(X=[a,b], \mathbb{F}\in\{{\mathbb{R},\mathbb{C}}\}\text{,}\) \(A=\{{\text{polynomials }[a,b]\rightarrow \mathbb{F}}\}=\{{x\mapsto\sum_{n=0}^N a_nx^n}\,:\,{a_n\in \mathbb{F}, N\geq 0}\}\) then \(A\) is a unital subalgebra of \(C(X,\mathbb{F})\text{.}\)
  • \(A_0=\{{\text{polynomials }p:[0,1]\rightarrow \mathbb{F}}\,:\,{p(0)=0}\}=\{{\sum_{n=1}^Na_nx^n}\,:\,{a_n\in F, N\geq 1}\}\) then \(A_0\) is a self-adjoint subalgebra of \(C(X,\mathbb{F})\) whic is not unital.
  • Let \(X=\mathbb{T}=\{{z\in \mathbb{C}}\,:\,{\left|{z}\right|=1}\}, \mathbb{F}=\mathbb{C}\text{.}\) Let \(A=\{{\text{polynomials }\mathbb{T}\rightarrow\mathbb{C}}\}=\{{z\mapsto\sum_{n=0}^Na_nz^n}\,:\,{a_n\in \mathbb{C}, N\geq 0}\}\text{.}\) Then \(A\) is a unital subalgebra of \(C(\mathbb{T})\text{.}\) Claim: \(A\) is not self adjoint.
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