June 11, 2026
We investigate two problems concerning spaceability in function spaces. First, we study the spaceability of sets of nowhere regular periodic functions, for various notions of regularity, including Hölder regularity and real analyticity. Second, we show that the set of real analytic functions is spaceable in \(C((0,1))\). In fact, we provide a complete characterization of the closed subspaces of \(C((0,1))\) consisting of real analytic functions. Our work solves several open questions posed by Bernal-González et al. [1]–[3].
A recurrent phenomenon in analysis is that sets of functions exhibiting a certain irregular behaviour are large, both in the topological and algebraic sense. For example, the set of continuous nowhere differentiable functions on \([0,1]\) is comeager in \(C([0,1])\) and contains an infinite-dimensional linear subspace, except for zero. The same holds for the set of smooth nowhere real analytic functions on \([0,1]\) within \(C^\infty([0,1])\). We refer to [4] for these and related results.
In this article, we are interested in the following notion of largeness: A subset \(M\) of a topological vector space \(E\) is said to be spaceable in \(E\) if \(M \cup \{0\}\) contains a closed infinite-dimensional subspace of \(E\). This terminology was coined by Gurariy and Quarta [5]. We study two problems related to spaceability in function spaces.
A classical result of Fonf, Gurariy, and Kadets [6] says that the set of continuous nowhere differentiable functions on \([0,1]\) is spaceable in \(C([0,1])\); see [7]–[10] for improvements and related results. Here, we study the spaceability of various sets of nowhere regular periodic functions. We consider Hölder regularity and ultradifferentiable classes, including real analyticity, and work with spaces of periodic functions whose spectrum is contained in a prescribed set \(\Gamma \subseteq \mathbb{Z}\), which includes spaces of disk algebra type (\(\Gamma = \mathbb{N}\)) as an important particular instance.
In the Hölder setting, we answer several questions posed by Bernal-González et al.in [2] and [3] about the spaceability of sets of functions in the disk algebra and holomorphic Hölder spaces on the unit disk \(\mathbb{D}\) whose restriction to \(\mathbb{T}= \partial \mathbb{D}\) is nowhere regular in a certain sense. In the real analytic case, we show that the set of functions in the smooth disk algebra \(A^\infty(\mathbb{D})\) whose restriction to \(\mathbb{T}\) is nowhere analytic is spaceable in \(A^\infty(\mathbb{D})\). This is related to the question on [3], asking whether the smaller set of functions in \(A^\infty(\mathbb{D})\) whose restriction to \(\mathbb{T}\) has a Pringsheim singularity at every point of \(\mathbb{T}\) is spaceable in \(A^\infty(\mathbb{D})\). Note that our result implies that the set of smooth nowhere real analytic functions on \([0,1]\) is spaceable in \(C^\infty([0,1])\). For general ultradifferentiable classes, our work complements the genericity results from [11], where spaceability is not considered. Moreover, unlike [11], we also cover the quasianalytic case.
Our approach combines properties of lacunary Fourier series [12], [13], expressing the principle that such series have the same regularity everywhere, together with a new general spaceability result for sets of periodic functions.
Gurariy [14] proved that the set of differentiable functions is not spaceable in \(C([0,1])\); see the introduction of [10] for an overview of related results. In contrast, Bernal-González [1] showed that the set of smooth functions is spaceable in \(C((0,1))\), and asked [1] whether the same holds for the smaller set of real analytic functions. We give a positive answer to this question and, in fact, obtain a complete characterization of the closed subspaces of \(C((0,1))\) consisting of real analytic functions. Furthermore, we show that every complemented subspace of \(C((0,1))\) consisting of real analytic functions is finite-dimensional.
Domański and Langenbruch [15] characterized the Fréchet subspaces of the space of real analytic functions on an open set \(\Omega \subseteq \mathbb{R}^d\) by using composition operators and abstract functional analysis. We rely heavily on this characterization for \(\Omega = (0,1)\) (and its proof). Bernal-González’s original approach is different and uses the theory of Müntz spaces [16]. We also indicate how the spaceability of the set of real analytic functions in \(C((0,1))\) can be obtained via this way.
In this preliminary section, we introduce the function spaces that will be used throughout the article.
We denote by \(L^1_{2\pi}\) the Banach space (of equivalence classes) of \(2\pi\)-periodic measurable functions \(f: \mathbb{R}\to \mathbb{C}\) such that \[\|f\|_{L^1_{2\pi}} = \frac{1}{2\pi} \int_{-\pi}^{\pi} |f(t)| {\rm d} t < \infty.\]
We define the Fourier coefficients of an element \(f \in L^1_{2\pi}\) as \[\widehat{f}(n) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(t) e^{-int} {\rm d} t, \qquad n \in \mathbb{Z}.\]
Let \(E\) be a topological vector space that is continuously included in \(L^1_{2\pi}\). For \(\Gamma \subseteq \mathbb{Z}\), we write \(E_\Gamma\) for the closed subspace of \(E\) consisting of all \(f \in E\) whose spectrum is contained in \(\Gamma\), i.e., \(\{ n\in \mathbb{Z}\mid \widehat{f}(n) \neq 0 \} \subseteq \Gamma\).
Let \(a = (a_n)_{n \in \mathbb{Z}}\) be a sequence of positive numbers with \(a_n \to 0\) as \(|n| \to \infty\). We denote by \(\mathcal{F}\ell^\infty_a\) the space consisting of all \(f \in L^1_{2\pi}\) such that \[|f|_{\mathcal{F}\ell^\infty_a} = \sup_{n \in \mathbb{Z}} \frac{|\widehat{f}(n)|}{a_n} < \infty.\] We endow \(\mathcal{F}\ell^\infty_a\) with the norm \[\|f\|_{\mathcal{F}\ell^\infty_a} = \|f\|_{L^1_{2\pi}} + |f|_{\mathcal{F}\ell^\infty_a}, \qquad f \in \mathcal{F}\ell^\infty_a.\] Then, \(\mathcal{F}\ell^\infty_a\) is a Banach space that is continuously included in \(L^1_{2\pi}\). We define \(\mathcal{F}c_{0,a}\) as the space consisting of all \(f \in L^1_{2\pi}\) such that \(\widehat{f}(n) = o(a_n)\) as \(|n| \to \infty\). Note that \(\mathcal{F}c_{0,a}\) is a closed subspace of \(\mathcal{F}\ell^\infty_a\).
We denote by \(C_{2\pi}\) the Banach space of \(2\pi\)-periodic continuous functions on \(\mathbb{R}\).
Let \(\alpha \in (0,1)\). The \(\alpha\)-Hölder space \(\Lambda^\alpha_{2\pi}\) consists of all \(f \in C_{2\pi}\) such that \[|f|_{\Lambda^\alpha_{2\pi}} = \sup_{\substack{t \in [-\pi,\pi] \\ 0< |h| \leq 1}} \frac{|f(t+h) - f(t)|}{|h|^\alpha} < \infty.\] We endow \(\Lambda^\alpha_{2\pi}\) with the norm \[\|f\|_{\Lambda^\alpha_{2\pi}} = |f(0)| + |f|_{\Lambda^\alpha_{2\pi}}, \qquad f \in \Lambda^\alpha_{2\pi}.\] Then, \(\Lambda^\alpha_{2\pi}\) is a Banach space that is continuously included in \(C_{2\pi}\). The little \(\alpha\)-Hölder space \(\lambda^\alpha_{2\pi}\) consists of all \(f \in C_{2\pi}\) such that \(f(t+h) - f(t) = o(|h|^\alpha)\) as \(|h| \to 0\) uniformly for \(t \in [-\pi,\pi]\). Note that \(\lambda^\alpha_{2\pi}\) is a closed subspace of \(\Lambda^\alpha_{2\pi}\).
Next, we introduce pointwise Hölder and Lipschitz conditions. Let \(\alpha \in (0,1)\) and \(t_0 \in \mathbb{R}\). A function \(f: \mathbb{R}\to \mathbb{C}\) is said to belong to \(\Lambda^\alpha(t_0)\) if \(f(t_0+h) - f(t_0) = O(|h|^\alpha)\) as \(h \to 0\), and to \(\lambda^\alpha(t_0)\) if \(f(t_0+h) - f(t_0) = o(|h|^\alpha)\) as \(h \to 0\). Similarly, we write \(f \in \operatorname{Lip}(t_0)\) if \(f(t_0+h) - f(t_0) = O(|h|)\) as \(h \to 0\).
We denote by \(C^\infty_{2\pi}\) the Fréchet space of \(2\pi\)-periodic smooth functions on \(\mathbb{R}\). We have the following characterization of \(C^\infty_{2\pi}\) in terms of Fourier coefficients:
Proposition 1. \(\displaystyle C^\infty_{2\pi} = \bigcap_{k>0} \mathcal{F}\ell^\infty_{(|n|^{-k})}\).
Next, we introduce ultradifferentiable classes defined via weight sequences; see [17] for more information. By a weight sequence, we mean here a sequence \(M = (M_p)_{p \in \mathbb{N}}\) of positive numbers with \(M^{1/p}_p \to \infty\) as \(p \to \infty\) that satisfies the following two conditions:
\(M\) is log-convex: \[M^2_{p} \leq M_{p-1}M_{p+1}, \qquad \forall p \in \mathbb{N}\backslash \{0\}.\]
\(M\) is shift-stable: There is \(H >0\) such that \[M_{p+1} \leq H^{p+1} M_{p}, \qquad \forall p \in \mathbb{N}.\]
Examples of weight sequences are the Gevrey sequences \((p!^s)\) with \(s >0\).
The associated function \(\omega_M\) of a weight sequence \(M\) is defined as \[\omega_M(\rho) = \sup_{p \in \mathbb{N}} \log \frac{M_0\rho^p}{M_p}, \qquad \rho > 0,\] and \(\omega_M(0) = 0\). Since \(M^{1/p}_p \to \infty\) as \(p \to \infty\), it holds that that \(\omega_M(t) < \infty\) for all \(t \geq 0\). Furthermore, \(\log t = o(\omega_M(t))\) as \(t \to \infty\).
Let \(M\) and \(N\) be two weight sequences. We write \(M \prec N\) if for every \(h >0\) there is \(C>0\) such that \[M_p \leq Ch^p N_p, \qquad \forall p \in \mathbb{N}.\]
Let \(M\) be a weight sequence. Given a compact interval \(J \subseteq \mathbb{R}\) and \(h >0\), we write \(\mathcal{E}^{M,h}(J)\) for the Banach space consisting of all \(f \in C^\infty(J)\) such that \[\| f \|_{\mathcal{E}^{M,h}(J)} = \sup_{p \in \mathbb{N}}\sup_{t \in J} \frac{|f^{(p)}(t)|}{h^pM_{p}} < \infty.\] Let \(I \subseteq \mathbb{R}\) be an open interval. We define the spaces of ultradifferentiable functions of Roumieu type \(\{M\}\) and Beurling type \((M)\) on \(I\) as \[\mathcal{E}^{\{M\}}(I) = \varprojlim_{J \Subset I} \varinjlim_{h \to \infty}\mathcal{E}^{M,h}(J), \qquad \mathcal{E}^{(M)}(I) = \varprojlim_{J \Subset I} \varprojlim_{h \to 0^+}\mathcal{E}^{M,h}(J),\] where \(J \Subset I\) means that \(J\) is a compact subinterval of \(I\). Note that \(\mathcal{E}^{\{p!\}}(I)\) is equal to the space \(\mathcal{A}(I)\) of real analytic functions on \(I\). We use \(\mathcal{E}^{[M]}(I)\) as a common notation for \(\mathcal{E}^{\{M\}}(I)\) and \(\mathcal{E}^{(M)}(I)\); a similar convention will be used for other spaces as well.
Let \(M\) and \(N\) be two weight sequences. If \(M \prec N\), then \(\mathcal{E}^{\{M\}}(I) \subseteq \mathcal{E}^{(N)}(I)\) with continuous inclusion.
Let \(M\) be a weight sequence. We denote by \(\mathcal{E}^{[M]}_{2\pi}\) the closed subspace of \(\mathcal{E}^{[M]}(\mathbb{R})\) consisting of \(2\pi\)-periodic functions. We define \[\mathcal{F}\ell^\infty_{\{\omega_M\}} = \bigcup_{h>0} \mathcal{F}\ell^\infty_{(e^{-\omega_M(|n|/h)})}, \qquad \mathcal{F}\ell^\infty_{(\omega_M)} = \bigcap_{h>0} \mathcal{F}\ell^\infty_{(e^{-\omega_M(|n|/h)})}.\] We have the following characterization of \(\mathcal{E}^{[M]}_{2\pi}\) in terms of Fourier coefficients (see e.g.[18]1):
Proposition 2. Let \(M\) be a weight sequence. Then, \(\mathcal{E}^{[M]}_{2\pi} = \mathcal{F}\ell^\infty_{[\omega_M]}\).
We denote by \(\mathbb{D}\) the open unit disk in \(\mathbb{C}\) and by \(\mathbb{T}\) its boundary. For \(f: \mathbb{T}\to \mathbb{C}\) we define the \(2\pi\)-periodic function \(\widetilde{f}(t) = f(e^{it})\), \(t \in \mathbb{R}\).
The disk algebra \(A(\mathbb{D})\) is the Banach space of all continuous functions on \(\overline{\mathbb{D}}\) that are analytic on \(\mathbb{D}\). We have the following characterization of \(A(\mathbb{D})\) in terms of Fourier coefficients:
Proposition 3. The map \(A(\mathbb{D}) \to C_{2\pi,\mathbb{N}}, \, f \mapsto \widetilde{f_{\mid \mathbb{T}}}\), is an isometric isomorphism.
Let \(\alpha \in (0,1)\). The holomorphic \(\alpha\)-Hölder space \(\Lambda^\alpha(\mathbb{D})\) consists of all \(f \in A(\mathbb{D})\) such that \[|f|_{\Lambda^\alpha(\mathbb{D})} = \sup_{\substack{z,w \in \mathbb{D}\\ z \neq w}} \frac{|f(z) - f(w)|}{|z-w|^\alpha} < \infty.\] We endow \(\Lambda^\alpha(\mathbb{D})\) with the norm \[\|f\|_{\Lambda^\alpha(\mathbb{D})} = |f(0)| + |f|_{\Lambda^\alpha(\mathbb{D})}, \qquad f \in \Lambda^\alpha(\mathbb{D}).\] Then, \(\Lambda^\alpha(\mathbb{D})\) is a Banach space that is continuously included in \(A(\mathbb{D})\). The holomorphic little \(\alpha\)-Hölder space \(\lambda^\alpha(\mathbb{D})\) consists of all \(f \in A(\mathbb{D})\) such that \(f(z) - f(w) = o(|z-w|^\alpha)\) as \(|z-w| \to 0\) uniformly for \(z,w \in \mathbb{D}\). Note that \(\lambda^\alpha(\mathbb{D})\) is a closed subspace of \(\Lambda^\alpha(\mathbb{D})\). We have the following characterization of \(\Lambda^\alpha(\mathbb{D})\) and \(\lambda^\alpha(\mathbb{D})\) in terms of Fourier coefficients:
Proposition 4. Let \(\alpha \in (0,1)\).
[19] The map \(\Lambda^\alpha(\mathbb{D}) \to \Lambda^\alpha_{2\pi,\mathbb{N}}, \, f \mapsto \widetilde{f_{\mid \mathbb{T}}}\), is a topological isomorphism.
[20] The map \(\lambda^\alpha(\mathbb{D}) \to \lambda^\alpha_{2\pi,\mathbb{N}}, \, f \mapsto \widetilde{f_{\mid \mathbb{T}}}\), is a topological isomorphism.
Next, we discuss pointwise Hölder and Lipschitz conditions. Let \(\alpha \in (0,1)\) and \(z_0 \in \mathbb{T}\). A function \(f: \mathbb{T}\to \mathbb{C}\) is said to belong to \(\Lambda^\alpha(z_0)\) if \(f(z) - f(z_0) = O(|z-z_0|^\alpha)\) as \(z \to z_0\), \(z \in \mathbb{T}\). The spaces \(\lambda^\alpha(z_0)\) and \(\operatorname{Lip}(z_0)\) are defined similarly.
Remark 5. Given \(\alpha \in (0,1)\) and \(t_0 \in \mathbb{R}\), it is clear that a function \(f: \mathbb{T}\to \mathbb{C}\) belongs to \(\Lambda^\alpha(e^{it_0})\) if and only if \(\widetilde{f} \in \Lambda^\alpha(t_0)\). A similar result holds for the pointwise \(\lambda^\alpha\)- and \(\operatorname{Lip}\)-spaces.
The smooth disk algebra \(A^\infty(\mathbb{D})\) is the Fréchet space of all smooth functions on \(\overline{\mathbb{D}}\) that are analytic on \(\mathbb{D}\). We have the following characterization of \(A^\infty(\mathbb{D})\) in terms of Fourier coefficients:
Proposition 6. The map \(A^\infty(\mathbb{D}) \to C^\infty_{2\pi,\mathbb{N}}, \, f \mapsto \widetilde{f_{\mid \mathbb{T}}}\), is a topological isomorphism.
In this section, we establish the spaceability of various sets of nowhere regular periodic functions. We consider Hölder and Lipschitz regularity and ultradifferentiable classes, including real analyticity.
We start with showing a general spaceability result involving the spaces \(\mathcal{F}\ell^\infty_{a}\) and \(\mathcal{F}c_{0,a}\).
Proposition 7. Let \(a = (a_n)_{n \in \mathbb{Z}}\) be a sequence of positive numbers with \(a_n \to 0\) as \(|n| \to \infty\) and \(\Gamma \subseteq \mathbb{Z}\) be infinite. Let \(E\) be a topolgical vector space such that \(E\) is continuously included in \(L^1_{2\pi}\) and \(\mathcal{F}\ell^\infty_{a,\Gamma} \subseteq E_{\Gamma}\). Then, the set \(E_\Gamma \backslash \mathcal{F}c_{0,a}\) is spaceable in \(E_\Gamma\).
Proof. Choose sequences \(\gamma_k = (\gamma_{k,j})_{j \in \mathbb{N}} \subseteq \Gamma\), \(k \in \mathbb{N}\), such that \[\{\gamma_{k,j} \mid j \in \mathbb{N}\} \cap \{\gamma_{k',j} \mid j \in \mathbb{N}\} =\emptyset, \qquad k\neq k',\] and \[\sum_{j = 0}^\infty a_{\gamma_{k,j}} < \infty.\] Define \[f_k(t) = \sum_{j = 0}^\infty a_{\gamma_{k,j}} e^{i\gamma_{k,j}t}, \qquad t \in \mathbb{R}.\] Then, \(f_k \in C_{2\pi}\) and, for \(n \in \mathbb{Z}\), \[\widehat{f}_k(n) = \left\{ \begin{array}{ll} a_{\gamma_{k,j}}, &ifn = \gamma_{k,j}for somej \in \mathbb{N},\\ 0, &otherwise. \end{array} \right.\] We obtain that \(f_k \in \mathcal{F}\ell^\infty_{a,\Gamma} \subseteq E\). Define \[X = \overline{\operatorname{span} \{f_k \mid k \in \mathbb{N}\}}^{E}.\] Note that \(X\) is infinite-dimensional as the family \(\{f_k \mid k \in \mathbb{N}\}\) is linearly independent. Let \(f \in X\). Then, \[f = \lim_{N \to \infty} \sum_{m=0}^{\nu_N} \alpha_{m,N} f_m \, \,in E,\] for certain \(\nu_N \in \mathbb{N}\) and \(\alpha_{m,N} \in \mathbb{C}\). As \(E\) is continuously included in \(L^1_{2\pi}\), we find that, for \(n \in \mathbb{Z}\), \[\widehat{f}(n) = \lim_{N \to \infty} \sum_{m=0}^{\nu_N} \alpha_{m,N} \widehat{f}_m(n) = \left\{ \begin{array}{ll} \alpha_k a_{\gamma_{k,j}}, & ifn = \gamma_{k,j}for somek,j \in \mathbb{N},\\ 0, &otherwise, \end{array} \right.\] where \(\alpha_k = \lim_{N \to \infty} \alpha_{k,N} \in \mathbb{C}\). Hence, \(f \in E_\Gamma\). If \(f \neq 0\), there exists \(k_0 \in \mathbb{N}\) such that \(\alpha_{k_0} \neq 0\) (otherwise, \(\widehat{f}(n) =0\) for all \(n \in \mathbb{Z}\) and thus \(f= 0\)). We obtain that, for all \(j \in \mathbb{N}\), \[\frac{|\widehat{f}(\gamma_{k_0,j})|}{a_{\gamma_{k_0,j}}} = | \alpha_{k_0}|.\] Consequently, \(f \notin \mathcal{F}c_{0,a}\). Thus, \(X\) is an infinite-dimensional closed subspace of \(E_\Gamma\) with \(X \backslash \{0\} \subseteq E_\Gamma \backslash \mathcal{F}c_{0,a}\). This shows the result. ◻
A set \(\Gamma \subseteq \mathbb{Z}\) is called (Hadamard) lacunary if \(\Gamma = (-\Gamma_1) \cup \Gamma_2\) with \(\Gamma_k \subseteq \mathbb{N}\), \(k= 1,2\), either finite or \(\Gamma_k = \{ \gamma_{k,j} \mid j \in \mathbb{N}\}\) such that \[\inf_{j \in \mathbb{N}} \frac{\gamma_{k,j+1}}{\gamma_{k,j}} > 1.\] As a rule of thumb, for periodic functions \(f\) whose spectrum is contained in a lacunary set, the regularity of \(f\) in the neigborhood of any point determines its global regularity. For Hölder and Lipschitz regularity, this may be expressed as follows (cf.[12], [13]):
Proposition 8. Let \(\Gamma \subseteq \mathbb{Z}\) be lacunary.
Let \(\alpha \in (0,1)\). For \(f \in L^1_{2\pi,\Gamma}\) the following statements are equivalent:
\(f \in \Lambda^\alpha_{2\pi}\) (\(f \in \lambda^\alpha_{2\pi}\)).
There is \(t_0 \in \mathbb{R}\) such that \(f \in \Lambda^\alpha(t_0)\) (\(f \in \lambda^\alpha(t_0)\)).
\(f \in \mathcal{F}\ell^\infty_{(|n|^{-\alpha})}\) ( \(f \in \mathcal{F}c_{0,(|n|^{-\alpha})}\)).
Let \(t_0 \in \mathbb{R}\). If \(f \in \operatorname{Lip}(t_0)\), then \(f \in \mathcal{F}\ell^\infty_{(|n|^{-1})}\).
Next, we consider ultradifferentiability. We believe the following result is known, but we give a proof for the sake of completeness.
Proposition 9. Let \(M\) be a weight sequence, \(\Gamma \subseteq \mathbb{Z}\) be lacunary, and \(f \in C_{2\pi,\Gamma}\). If \(f_{\mid I}\in \mathcal{E}^{[M]}(I)\) for some non-empty open interval \(I \subseteq \mathbb{R}\), then \(f \in \mathcal{E}^{[M]}_{2\pi}\).
Proof. We may assume that \(I \subseteq (-\pi,\pi)\). Fix a non-empty compact subinterval \(J\) of \(I\). Since \(\Gamma\) is lacunary, [13] implies that there is \(C >0\) such that for all \(f \in C_{2\pi,\Gamma}\) and \(n \in \mathbb{Z}\) \[|\widehat{f}(n)| \leq C \left(\int_{J} |f(t)|^2 {\rm d} t \right)^{1/2}.\] Let \(h >0\) and \(f \in C_{2\pi,\Gamma}\) be such that \(f_{\mid I} \in \mathcal{E}^{M,h}(I)\). Then, for all \(p \in \mathbb{N}\) and \(n \in \mathbb{Z}\), \[|n|^p|\widehat{f}(n)| = | \widehat{f^{(p)}}(n)| \leq C \left(\int_{J} |f^{(p)}(t)|^2 {\rm d} t \right)^{1/2} \leq C|J|^{1/2} \| f \|_{\mathcal{E}^{M,h}(J)} h^p M^p.\] Hence, \[|\widehat{f}(n)| \leq C|J|^{1/2} \| f \|_{\mathcal{E}^{M,h}(J)} \inf_{p \in \mathbb{N}} \frac{ h^p M_p}{|n|^p} = M_0C|J|^{1/2} \| f \|_{\mathcal{E}^{M,h}(J)} e^{-\omega_M(|n|/h)}.\] The result now follows from Proposition 2. ◻
In the remainder of this section, we combine Proposition 7 with the above two results to obtain several spaceability results about nowhere regular periodic functions.
We start with a result about continuous nowhere Hölder regular periodic functions.
Theorem 10. Let \(\Gamma \subseteq \mathbb{Z}\) be infinite. The set \[\label{set2} \{f \in C_{2\pi,\Gamma} \mid f \notin \Lambda^\alpha(t)for allt \in \mathbb{R} and\alpha \in (0,1)\}\tag{1}\] is spaceable in \(C_{2\pi, \Gamma}\).
Proof. Define \(a = (1/\log(e+|n|))\). We may assume that \(\Gamma\) is lacunary and that \(\sum_{\gamma \in \Gamma} a_\gamma < \infty\) (otherwise, replace \(\Gamma\) by a subset of \(\Gamma\) that satisfies these properties). Then, \(\mathcal{F}\ell^\infty_{a,\Gamma} \subseteq C_{2\pi,\Gamma}\). Since \(|n|^{-\alpha}= o(a_n)\) as \(|n| \to \infty\) for all \(\alpha \in (0,1)\), we obtain that \[\bigcup_{\alpha \in (0,1)} \mathcal{F}\ell^\infty_{(|n|^{-\alpha})} \subseteq \mathcal{F}c_{0,a}.\] By Proposition 7, \(C_{2\pi,\Gamma} \backslash \mathcal{F}c_{0,a}\) is spaceable in \(C_{2\pi,\Gamma}\). Hence, also the bigger set \[C_{2\pi,\Gamma} \backslash \bigcup_{\alpha \in (0,1)} \mathcal{F}\ell^\infty_{(|n|^{-\alpha})}\] is spaceable in \(C_{2\pi,\Gamma}\). Proposition 8 gives that the latter set coincides with the one in 1 . ◻
The following result gives a positive answer to the question posed on [3].
Corollary 1. The set \[\{f \in A(\mathbb{D}) \mid f_{\mid \mathbb{T}} \notin \Lambda^\alpha(z)for allz \in \mathbb{T} and\alpha \in (0,1)\}\] is spaceable in \(A(\mathbb{D})\).
Theorem 11. Let \(\alpha \in (0,1)\) and \(\Gamma \subseteq \mathbb{Z}\) be infinite. The set \[\{f \in \lambda^\alpha_{2\pi,\Gamma} \mid f \notin \Lambda^\beta(t)for allt \in \mathbb{R} and\beta \in (\alpha,1)\}\] is spaceable in \(\lambda^\alpha_{2\pi, \Gamma}\).
Proof. We may assume that \(\Gamma\) is lacunary. Proposition 8 implies that \(\lambda^\alpha_{2\pi,\Gamma} = \mathcal{F}c_{0,(|n|^{-\alpha}),\Gamma}\). Define \(a = (|n|^{-\alpha}/\log(e+|n|))\). Since \(a_n = o(|n|^{-\alpha})\) and \(|n|^{-\beta}= o(a_n)\) as \(|n| \to \infty\) for all \(\beta \in (\alpha,1)\), we obtain that \[\bigcup_{\beta \in (\alpha,1)} \mathcal{F}\ell^\infty_{(|n|^{-\beta})} \subseteq \mathcal{F}c_{0,a} \subseteq \mathcal{F}\ell^\infty_{a} \subseteq \mathcal{F}c_{0,(|n|^{-\alpha})}.\] The remainder of the proof is similar to the one of Theorem 10. ◻
Corollary 2. Let \(\alpha \in (0,1)\). The set \[\{f \in \lambda^\alpha(\mathbb{D}) \mid f_{\mid \mathbb{T}} \notin \Lambda^\alpha(z)for allz \in \mathbb{T} and\beta \in (\alpha,1) \}\] is spaceable in \(\lambda^\alpha(\mathbb{D})\).
Remark 12. Let \(\alpha \in (0,1)\). Corollary 2 implies that the set \[\{f \in \lambda^\alpha(\mathbb{D}) \mid f_{\mid \mathbb{T}} \notin \operatorname{Lip}(z)for allz \in \mathbb{T}\}\] is spaceable in \(\lambda^\alpha(\mathbb{D})\). This gives a positive answer to the first part of question (a) posed on [2].
For \(\alpha \in (0,1]\) we define the Fréchet space \[\widetilde{\Lambda}^\alpha_{2\pi} = \varprojlim_{\beta \to \alpha-} \Lambda^\beta_{2\pi}.\]
Theorem 13. Let \(\Gamma \subseteq \mathbb{Z}\) be infinite.
Let \(\alpha \in (0,1)\). The set \[\{f \in \widetilde{\Lambda}^\alpha_{2\pi,\Gamma} \mid f \notin \Lambda^\alpha(t)for allt \in \mathbb{R}\}\] is spaceable in \(\widetilde{\Lambda}^\alpha_{2\pi, \Gamma}\).
The set \[\{f \in \widetilde{\Lambda}^1_{2\pi,\Gamma} \mid f \notin \operatorname{Lip}(t)for allt \in \mathbb{R}\}\] is spaceable in \(\widetilde{\Lambda}^1_{2\pi, \Gamma}\).
Proof. Let \(\alpha \in (0,1]\) (we prove (i) and (ii) simultaneously). We may assume that \(\Gamma\) is lacunary. Proposition 8 yields that \[\widetilde{\Lambda}^\alpha_{2\pi,\Gamma} = \bigcap_{\beta \in (0,\alpha)}\mathcal{F}\ell^\infty_{(|n|^{-\beta}),\Gamma}.\] Define \(a = (|n|^{-\alpha}\log(e+|n|))\). Since \(a_n = O(|n|^{-\beta})\) and \(|n|^{-\alpha}= o(a_n)\) as \(|n| \to \infty\) for all \(\beta \in (0,\alpha)\), we obtain that \[\mathcal{F}\ell^\infty_{(|n|^{-\alpha})} \subseteq \mathcal{F}c_{0,a} \subseteq \mathcal{F}\ell^\infty_{a} \subseteq \bigcap_{\beta \in (0,\alpha)} \mathcal{F}\ell^\infty_{(|n|^{-\beta})}.\] The remainder of the proof is similar to the one of Theorem 10. ◻
For \(\alpha \in (0,1]\) we define the Fréchet space \[\widetilde{\Lambda}^\alpha(\mathbb{D}) = \varprojlim_{\beta \to \alpha-} \Lambda^\beta(\mathbb{D}).\]
Corollary 3. Let \(\Gamma \subseteq \mathbb{Z}\) be infinite.
Let \(\alpha \in (0,1)\). The set \[\{f \in \widetilde{\Lambda}^\alpha(\mathbb{D}) \mid f_{\mid \mathbb{T}} \notin \Lambda^\alpha(z)for allz \in \mathbb{T}\}\] is spaceable in \(\widetilde{\Lambda}^\alpha(\mathbb{D})\).
The set \[\label{set3} \{f \in \widetilde{\Lambda}^1(\mathbb{D}) \mid f_{\mid \mathbb{T}} \notin \operatorname{Lip}(z)for allz \in \mathbb{T}\}\tag{2}\] is spaceable in \(\widetilde{\Lambda}^1(\mathbb{D})\).
Remark 14. The second part of Corollary 3 addresses question (b) on [2], which asked what lineability properties the set in 2 enjoys.
We start with the following technical lemma.
Lemma 1. Let \(M\) be a weight sequence. For every \(L >1\) it holds that \(\omega_M(L\rho) - \omega_M(\rho) \to \infty\) as \(\rho \to \infty\).
Proof. Define \(m_p = M_p/M_{p-1}\), \(p \geq 1\), and the counting function \(m(\lambda) = \sum_{m_p \leq \lambda} 1\), \(\lambda \geq 0\). By [17], we have the following representation of \(\omega_M\): \[\omega_M(\rho) = \int_0^\rho \frac{m(\lambda)}{\lambda} {\rm d} \lambda, \qquad \rho \geq 0.\] For \(L >1\) we obtain that \[\omega_M(L\rho) - \omega_M(\rho) = \int_{\rho}^{L\rho} \frac{m(\lambda)}{\lambda} {\rm d} \lambda \geq m(\rho)\log L,\] from which the result follows. ◻
Given a weight function \(M\), a function \(f: \mathbb{R}\to \mathbb{C}\) is said to be nowhere of class \(\mathcal{E}^{[M]}\) if \(f_{\mid I} \notin \mathcal{E}^{[M]}(I)\) for all non-empty open intervals \(I \subseteq \mathbb{R}\).
Theorem 15. Let \(M\) be a weight sequence and \(\Gamma \subseteq \mathbb{Z}\) be infinite. The set \[\label{set4} \{f \in C^\infty_{2\pi,\Gamma} \mid fis nowhere of class\mathcal{E}^{\{M\}}\}\tag{3}\] is spaceable in \(C^\infty_{2\pi,\Gamma}\).
Proof. We may assume that \(\Gamma\) is lacunary. Proposition 1 yields that \[C^\infty_{2\pi} = \bigcap_{k >0}\mathcal{F}\ell^\infty_{(|n|^{-k})}.\] Define \(a = (e^{-\omega_M\left(\sqrt{|n|}\right)})\). Note that \(a_n = O(|n|^{-k})\) as \(|n| \to \infty\) for all \(k >0\) and, by Lemma 1, \(e^{-\omega_M(|n|/h)} = o(a_n)\) as \(|n| \to \infty\) for all \(h >0\). Consequently, \[\mathcal{F}\ell^\infty_{\{\omega_M\}} \subseteq \mathcal{F}c_{0,a} \subseteq \mathcal{F}\ell^\infty_{a} \subseteq \bigcap_{k >0}\mathcal{F}\ell^\infty_{(|n|^{-k})}.\] By Proposition 7, \(C^\infty_{2\pi,\Gamma} \backslash \mathcal{F}c_{0,a}\) is spaceable in \(C^\infty_{2\pi,\Gamma}\). Hence, also the bigger set \(C^\infty_{2\pi,\Gamma} \backslash \mathcal{F}\ell^\infty_{\{\omega_M\}}\) is spaceable in \(C^\infty_{2\pi,\Gamma}\). Propositions 2 and 9 imply that the latter set coincides with the one in 3 . ◻
Corollary 4. The set \[\{f \in A^\infty(\mathbb{D}) \mid \widetilde{f_{\mid \mathbb{T}}}is nowhere real analytic on\mathbb{R}\}\] is spaceable in \(A^\infty(\mathbb{D})\).
Proof. This follows from Proposition 6 and Theorem 15 with \(M = (p!)\) and \(\Gamma = \mathbb{N}\). ◻
Remark 16. Corollary 4 sheds some light on the question on [3], which asked whether the smaller set \[\{f \in A^\infty(\mathbb{D}) \mid \widetilde{f_{\mid \mathbb{T}}}is Pringsheim singular at every point of\mathbb{R}\}\] is spaceable in \(A^\infty(\mathbb{D})\).
Theorem 17. Let \(M\) and \(N\) be two weight sequences with \(M \prec N\) and \(\Gamma \subseteq \mathbb{Z}\) be infinite. The set \[\{f \in \mathcal{E}^{(N)}_{2\pi, \Gamma} \mid fis nowhere of class\mathcal{E}^{\{M\}}\}\] is spaceable in \(\mathcal{E}^{(N)}_{2\pi, \Gamma}\).
Proof. We may assume that \(\Gamma\) is lacunary. Define \(Q = (\sqrt{M_pN_p})\). Then, \(Q\) is a weight sequence satisfying \(M \prec Q \prec N\). By [17], for every \(h > 0\) there are \(C_1,C_2 >0\) such that \[\label{wsmiddle} \omega_Q(\rho) \leq \omega_M(\rho/h) + C_1 \quad and \quad \omega_N(\rho/h) \leq \omega_Q(\rho) + C_2, \qquad \forall \rho \geq 0.\tag{4}\] Proposition 2 yields that \(\mathcal{E}^{(N)}_{2\pi} = \mathcal{F}\ell^\infty_{(\omega_N)}\). Define \(a = (e^{-\omega_Q(|n|)})\). By Lemma 1 and 4 , \(e^{-\omega_M(|n|/h)} = o(a_n)\) and \(a_n =O(e^{-\omega_N(|n|/h)})\) as \(|n| \to \infty\) for all \(h >0\). Consequently, \[\mathcal{F}\ell^\infty_{\{\omega_M\}} \subseteq \mathcal{F}c_{0,a} \subseteq \mathcal{F}\ell^\infty_{a} \subseteq \mathcal{F}\ell^\infty_{(\omega_N)}.\] The remainder of the proof is similar to the one of Theorem 15. ◻
Remark 18. Let \(M\) and \(N\) be two weight sequences with \(M \prec N\). Since \(\mathcal{E}^{(N)}_{2\pi}\) is a topological subspace of \(\mathcal{E}^{(N)}(\mathbb{R})\), Theorem 17 (with \(\Gamma = \mathbb{Z}\)) implies that the set \[\label{setrem} \{f \in \mathcal{E}^{(N)}(\mathbb{R}) \mid fis nowhere of class\mathcal{E}^{\{M\}}\}\tag{5}\] is spaceable in \(\mathcal{E}^{(N)}(\mathbb{R})\). This complements [11], in which it is shown that the set in 5 is \(\mathfrak{c}\)-dense-lineable \(\mathcal{E}^{(N)}(\mathbb{R})\), under the assumption that \(M\) is non-quasianalytic.
Let \(I \subseteq \mathbb{R}\) be an open interval and write \(C(I)\) for the Fréchet space of continuous functions on \(I\). This section is devoted to the study of the closed subspaces of \(C(I)\) consisting of functions that are real analytic. In particular, we show that the set \(\mathcal{A}(I)\) is spaceable in \(C(I)\). As mentioned in the introduction, we rely heavily on the work [15] of Domański and Langenbruch.
We denote by \(\mathcal{O}(\mathbb{D})\) the Fréchet space of analytic functions on \(\mathbb{D}\). The following result is a direct consequence of the proof of [15].
Proposition 19. Let \(I \subseteq \mathbb{R}\) be a non-empty open interval. There exists a real analytic function \(\varphi: I \to \mathbb{D}\) such that the composition operator \[\mathcal{O}(\mathbb{D}) \to C(I), \, f \mapsto f \circ \varphi,\] is a topological embedding.
Proposition 19 implies the following result, which gives a positive answer to [1].
Corollary 5. Let \(I \subseteq \mathbb{R}\) be a non-empty open interval. The set \(\mathcal{A}(I)\) is spaceable in \(C(I)\).
In the next remark, we indicate an alternative proof of Corollary 5 (with \(I = (0,1)\)) by means of the theory of Müntz spaces [16]. This approach was used by Bernal-González in [1] to show that the set \(C^\infty((0,1))\) is spaceable in \(C((0,1))\).
Remark 20. Let \((\lambda_n)_{n \in \mathbb{N}}\) be an increasing sequence of positive numbers that satisfies \(\inf_{n \in \mathbb{N}} \lambda_{n+1} - \lambda_n >0\) and \(\sum_{n = 1}^\infty 1/\lambda_n < \infty\). Define \[E= \overline{ \operatorname{span} \{ t^{\lambda_n} \mid n \in \mathbb{N}\}}^{C((0,1))}.\] Clearly, \(E\) is an infinite-dimensional closed subspace of \(C((0,1))\). By using the same argument as in [16], but using the coefficient bounds from [16] instead of [16], we see that every function in \(E\) is real analytic on \((0,1)\).
We now give a characterization of the closed subspaces of \(C(I)\) consisting of real analytic functions.
Theorem 21. Let \(I \subseteq \mathbb{R}\) be a non-empty open interval. A Fréchet space is isomorphic to a closed subspace of \(C(I)\) consisting of real analytic functions on \(I\) if and only if it is isomorphic to a closed subspace of \(\mathcal{H}(\mathbb{D})\).
Proof. Proposition 19 implies that any Fréchet space that is isomorphic to a closed subspace of \(\mathcal{H}(\mathbb{D})\) is also isomorphic to a closed subspace of \(C(I)\) consisting of real analytic functions on \(I\). Conversely, let \(E\) be a Fréchet space that is isomorphic to a closed subspace of \(C(I)\) consisting of real analytic functions on \(I\). In [15] it is shown that every closed Fréchet subspace of \(\mathcal{A}(I)\) is isomorphic to a closed subspace of \(\mathcal{H}(\mathbb{D})\). Hence, it suffices to show that \(E\) is isomorphic to a closed subspace of \(\mathcal{A}(I)\). There exists a topological embedding \(T: E \to C(I)\) with \(T(E) \subseteq \mathcal{A}(I)\). Note that \(\mathcal{A}(I)\) is webbed and \(E\) is ultrabornological. De Wilde’s closed graph theorem [21] therefore implies that the map \(T: E \to \mathcal{A}(I)\) is continuous. Since \(\mathcal{A}(I)\) is continuously included into \(C(I)\), we find that \(T: E \to \mathcal{A}(I)\) is a topological embedding, which yields the result. ◻
Remark 22. Note that \(\mathcal{H}(\mathbb{D}) \cong \Lambda_0(j)\) via Taylor coefficients; see [21] for the definition of power series spaces. We refer to [22] for an internal characterization of the closed subspaces of \(\Lambda_0(j)\).
In the next result, we show that \(C(I)\) does not contain infinite-dimensional complemented subspaces consisting of real analytic functions:
Proposition 23. Let \(I \subseteq \mathbb{R}\) be a non-empty open interval. Every complemented subspace of \(C(I)\) consisting of real analytic functions on \(I\) is finite-dimensional.
Proof. Let \(E\) be a complemented subspace of \(C(I)\) consisting of real analytic functions on \(I\). There exists a continuous linear projection \(P: C(I) \to C(I)\) onto \(E\). Fix a compact interval \(J_0 \subseteq I\) with non-empty interior. Choose a compact subinterval \(J \subseteq I\) and \(C >0\) such that \[\label{propp} \sup_{x \in J_0}|P(f)(x)| \leq C \sup_{x \in J} |f(x)|, \qquad \forall f \in C(I).\tag{6}\] We write \(\rho_J: C(I) \to C(J), \, f \mapsto f_{\mid J}\), for the restriction map. We claim that \(\rho_{J \mid E}: E \to C(J)\) is a topological embedding. As \(\rho_{J}(E) \subseteq \mathcal{A}(J) \subseteq C^1(J)\), the result would follow from the fact that the set of differentiable functions on \(J\) is not spaceable in \(C(J)\) [14]. We now show the claim. Since \(P(C(I)) = E \subseteq \mathcal{A}(I)\), 6 and the uniqueness property of real analytic functions imply that \(P(f) = 0\) for all \(f \in C(I)\) with \(f_{\mid J} = 0\). As the restriction \(\rho_J: C(I) \to C(J)\) is a quotient map and \(P = 0\) on \(\ker \rho_J\), there exists a continuous linear map \(\tilde{P}: C(J) \to C(I)\) such that \(\tilde{P} \circ \rho_J = P\). Then, \(\operatorname{id}_{E} = P_{\mid E} = \tilde{P} \circ \rho_{J \mid E}\), which implies the claim. ◻
Remark 24. Proposition 23 is related to the much deeper result [23] of Domański and Vogt that every complemented Fréchet subspace of \(\mathcal{A}(I)\) is finite-dimensional.
Remark 25. In contrast to Proposition 23, there do exist infinite-dimensional complemented subspaces of \(C(I)\) consisting of smooth functions. We may suppose that \(I = \mathbb{R}\). Pick \(\varphi_n \in C^\infty(\mathbb{R})\), \(n \in \mathbb{N}\), with \(\varphi_n(n) = 1\) and \(\operatorname{supp} \varphi_n \subseteq [n- \frac{1}{2}, n + \frac{1}{2}]\). Define \[E = \left \{ \sum_{n =0}^\infty \alpha_n \varphi_n \mid \alpha_n \in \mathbb{C} for alln \in \mathbb{N}\right \}.\] Then, \(E \subseteq C^\infty(\mathbb{R})\) and \(E\) is infinite-dimensional. The continuous linear map \[P: C(\mathbb{R}) \to C(\mathbb{R}), \, f \mapsto \sum_{n =0}^\infty f(n) \varphi_n,\] is a projection onto \(E\).
This extends [1], in which it is shown that \(C^\infty(I)\) is spaceable in \(C(I)\). In addition, it gives an alternative proof of this result.