Appendix B - Linear Analysis

Summary

In this appendix, we collect some miscellaneous topics from linear analysis referred throughout the main text.

Stone–Weierstrass Theorem

In this section, we present a proof of the Stone–Weiersrass theorem, which does not rely on the Weierstrass theorem. We closely follow the treatment of [119].

Throughout this section, K denotes a compact Hausdorff space.

Theorem B.1.1

Let be an algebra of continuous functions with the following properties:

• (1) If, then there exists such that.

• (2) For every, there exists such that.

Then, is dense in the algebra of continuous real-valued functions on K endowed with the uniform norm.

We start the proof with a lemma, which shows that under some modest assumption, pointwise convergence yields uniform convergence.

Lemma B.1.1

Let ﹛f_n﹜ be a sequence in C[a, b] converging pointwise to a continuous function f. If is decreasing for all, then converges uniformly to f.

Proof Let. For, consider the closed subset

of [a, b]. As,. In particular, finite intersection of sets from ﹛K_n﹜ is non-empty if every K_n is non-empty. If each K_n is non-empty, then by Cantor's intersection theorem,. However, if, then as, for sufficiently large n. Hence, K_N is empty for some N, that is, for every x ∈ [a, b] and for every.

Here is an important special case of Weierstrass’ theorem.

Lemma B.1.2

Define a sequence of polynomials by p₀(x) = 0, and

If q_n(x) = p_n(x²), then converges uniformly to f (x) = |x| on [−1, 1].

Proof A routine calculation shows that

One may now verify inductively that

In particular, converges pointwise to |x|. Now apply Lemma B.1.1 to.

The last lemma yields some basic properties of closed subalgebras of.

Lemma B.1.3

Let be a subalgebra of C_ℝ(K) and let denote the uniform closure of in C_ℝ(K)., then so are and.

Proof Recall that. It may be concluded from Lemma B.1.2 that for ever. The first part now is immediate from

and finite induction, whereas the remaining part follows from.