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2016 | OriginalPaper | Chapter

8. Spaces of Continuous Functions

Author : Vilmos Komornik

Published in: Lectures on Functional Analysis and the Lebesgue Integral

Publisher: Springer London

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Abstract

In this chapter the letter K always denotes a compact Hausdorff space. We recall from topology that the continuous functions $f: K \rightarrow \mathbb{R}$ form a Banach space C(K) with respect to the norm

$$\displaystyle{ \left \Vert \,f\right \Vert _{\infty }:=\max _{t\in K}\vert \,f(t)\vert, }$$

and that norm convergence is uniform convergence on K. We will only present some basic results.

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previous chapter Integrals on Measure Spaces

next chapter Spaces of Integrable Functions

Gillman–Jerison [169] and Semadeni [421] treat many further topics.

We recall that $\mathop{\mathrm{med}}\nolimits \left \{x,y,z\right \}$ denotes the middle number among x, y and z.

See Gelbaum–Olmsted [168]. The situation is similar to that of c ₀; see p. 140.

Weierstrass [483], p. 5.

Landau [283]. See Proposition 8.16 and Exercise 8.3 below (pp. 282,300) for other proofs.

ω( f, δ) is called the uniform continuity modulus of f.

See the references in the footnote of Sect. 9.3 below, p. 320.

We recall that in this book by a subspace without adjective we always mean a linear subspace.

Several proofs of this chapter could be simplified by adopting the complex framework, and using Euler’s formula $e^{ix} =\cos x + i\sin x$. For example, the trigonometric polynomials would be simply the algebraic polynomials of e ^it, and the single identity $e^{u+v} = e^{u}e^{v}$ would suffice instead of these three real identities.

Weierstrass [483]. See Theorem 8.11 and a remark following Proposition 8.21 below (pp. 276, 288) for other proofs.

de la Vallée-Poussin [463]. His work was motivated by that of Landau.

Stone [440], [441].

Kakutani [240, pp. 1004–1005], Krein–Krein [268].

See the proof of Lemma 8.27, p. 297.

Stone [441]. This is a version of similar theorems of Urysohn [461] and Tietze [453].

We have already used this technique when proving the Riesz Lemma 5.13, p. 184.

We recall that a set A is totally bounded or precompact if for each r > 0 it has a finite cover by balls of radius r.

Ascoli [12] (pp. 545–549, sufficiency for K = [0, 1]), Arzelà [8] (necessity), [9] (simplified treatment), [10], Fréchet [154] (general case).

We recall that the bounded and totally bounded sets are the same in all finite-dimensional normed spaces.

In this formula the balls are taken in $\mathbb{R}^{m}$.

We recall that r > 0 was chosen arbitrarily at the beginning.

Daniel Bernoulli [38], Fourier [148]. Using complex numbers the Fourier series would take the simpler form $\sum _{k=-\infty }^{\infty }c_{k}e^{ikx}$.

A fascinating historical account is given by Kahane [237].

Dirichlet [112], Jordan [229].

Lipschitz [308] and Dini [107], [110]. See a short proof in Exercise 8.5, p. 301.

du Bois-Reymond [49], [51]. A simpler explicit counterexample was given later by Fejér [139], [140]. We prove here the mere existence of such functions.

Carleson [78]. This was a long-standing open problem of Lusin [313]. See also the remark following Corollary 9.6 below (p. 314) concerning L ^p convergence.

Kahane and Katznelson [238]. See also Edwards [120], Katznelson [245] and Zygmund [493] for many further results.

Dirichlet [112].

For $\sin s = 0$ we replace the right-hand side by its limit (2m + 1).

Fejér [141]. See also Edwards [120] or Zygmund [493].

See Hawkins [198]. An analogous phenomenon for Taylor series has been known since Cauchy [80, p. 230].

Fejér [137, 138]. He also investigated pointwise convergence for discontinuous functions f. Lebesgue [292] extended his results to Lebesgue integrable functions.

Thereby he has answered Minkowski’s question. Banach [20] has shown that Minkowski’s phenomenon occurs for a slight modification of the trigonometric system.

Cauchy [79].

For $\sin s = 0$ the right-hand side is replaced by its limit (n + 1).

Korovkin [263]. Many applications are given in Korovkin [264].

Freud [153]. See Altomare and Campiti [5] for a very complete review of the subject.

Bohman [47], Korovkin [263].

Theorem 8.1, p. 260.

Bernstein’s proof is probabilistic, based on the law of large numbers.

Bernstein [39]. His result answered a question of Borel [60, pp. 79–82].

We denote by id the identity map of I.

Lozinski [311].

Marcinkiewicz [314], Lozinski [310].

The proof may be simplified by using complex numbers. See Exercise 8.10, p. 303.

Lozinski [311].

By a weight function we mean a positive, integrable function. If w is a weight function on a compact interval J, then we may define a scalar product on the vector space $\mathcal{P}$ of algebraic polynomials by the formula $(p,q):=\int _{I}pqw\ dt$, and we may apply the Gram–Schmidt orthogonalization (Proposition 1.15, p. 28) for the sequence of functions 1, id, id², …to obtain a sequence of orthogonal polynomials satisfying $\deg p_{k} = k$ for every k = 0, 1, ….

Nikolaev [346]. However, we will see later (Corollary 9.6, p. 314) that the answer is affirmative for the weaker norm associated with the scalar product.

Faber [133].

Fejér [142]; see also Cheney [85]. In this way, Hermite interpolation can be used to prove the Weierstrass approximation theorem.

Erdős–Turán [124].

Erdős–Vértesi [125].

Baire [17].

Here $\left \vert \mu \right \vert$ denotes the total variation of μ; see p. 231.

Riesz [377] (K = [0, 1]), Radon [366] ($K \subset \mathbb{R}^{N}$, p. 1333), Banach [25] and Saks [410] (compact metric spaces), Markov [315] (C _b(K) certain non-compact spaces), Kakutani [240] (compact topological spaces). See also the beautiful simple proof of Riesz for K = [0, 1]: Riesz and Sz.-Nagy [394, Sect. 50].

We will need it only during the proof of Lemma 8.27 below, in order to apply Proposition 8.6.

Dini [109, Sect. 99]. See the graphs of the functions f _n (t):= t ⁿ for n = 1, 2, 3 in Fig. 8.15, and let K = [0,a], 0 < a < 1.

Kindler [248].

See Fig. 8.16.

The proof is similar to that of ordinary intervals.

$\mathcal{M}$ is even a $\sigma$-algebra.

The finiteness follows from the relation $\mu (K) =\varphi (1) < \infty $.

We apply the preceding identity to f∕a and f∕b, and we take the differences of the resulting equalities.

We have g′(x) = 0 if f(x) ≥ h′(x), and $0 \leq g'(x) = h'(x) - f(x) \leq g(x)$ otherwise.

As usual, f ⁺ and f ⁻ denote the positive and negative parts of f.

If K is metrizable, then we may define f explicitly by the formula

$$\displaystyle{ f(t):= \frac{\mathop{\mathrm{dist}}\nolimits (t,N') -\mathop{\mathrm{dist}}\nolimits (t,P')} {\mathop{\mathrm{dist}}\nolimits (t,N') +\mathop{ \mathrm{dist}}\nolimits (t,P')}. }$$

We follow Riesz–Sz.-Nagy [394].

The second formula is meaningful because m has bounded variation and hence at most countably many discontinuities.

Komornik–Yamamoto [261, 262] apply such estimates to inverse problems.

Compare this with the proof of the non-separability of $\ell^{\infty }$, p. 74.

For the characterization of the weakly compact sets of C(K) see, e.g., Dunford–Schwartz [117].

Visser [469], see Sz.-Nagy [448, p. 77.].

Lebesgue [286, 296].

Chebyshev [83], Borel [60].

For brevity we use the complex notation.

Chernoff [86]. The method is quite general and leads to an improvement of the classical theorems of Lipschitz and Dini. It was motivated by an earlier simple proof of the Fourier inversion theorem by Richards [369].

The first examples were due to Bolzano [55] around 1832 (published only in 1930) and Weierstrass [480, 481]. See also Bolzano [57], Russ [407], Jarník [227, p. 37], du Bois-Reymond [50], Dini [108], Hawkins [198].

Takagi [449]. His example was rediscovered by van der Waerden [477]. See also Billingsley [44], Shidfar–Sabetfakhiri [422], McCarthy [319].

Peano [354]. The following proof is due to Lebesgue [297, pp. 44–45]. An interesting variant of this proof is due to Schoenberg [417]. See also Aleksandrov [4].

Schauder [412].

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Title: Spaces of Continuous Functions
Author: Vilmos Komornik
Publisher: Springer London
Book: Lectures on Functional Analysis and the Lebesgue Integral
Print ISBN: 978-1-4471-6810-2

Electronic ISBN: 978-1-4471-6811-9

Copyright Year: 2016
DOI: https://doi.org/10.1007/978-1-4471-6811-9_8

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