Abstract
This work is a contribution to the ongoing search for algebraic structures within a nonlinear setting. Here, we shall focus on the study of lineability of subsets of continuous functions on the one hand and within the setting of Sobolev spaces on the other (which represents a novelty in the area of research).
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1 Introduction and preliminaries
Since its appearance in 2005, the terminology lineability and spaceability has attracted the attention of many researchers and, just recently, the American Mathematical Society introduced this terminology in its 2020 Mathematical Subject Classification under the references 15A03 and 46B87. In a nutshell, this notion consists on finding (when possible) large algebraic structures within nonlinear subsets of a topological vector space.
Some early examples of results within this theory are due to V. I. Gurariy (1935–2005), who proved that the set of continuous nowhere differentiable functions on \(\mathbb {R}\) contains (except for \(\{0\}\)) infinite dimensional linear spaces. Furthermore, in 2005 [1], he proved that the set of differentiable nowhere monotone functions also contains (except for \(\{0\}\)) infinite dimensional linear spaces. After these seminal works, a lot has been done linking many areas of mathematics, such as Set Theory [8, 9, 11], Real and Complex Analysis [10, 18], Linear and Multilinear Algebra [5], Linear Dynamics [15], or Statistics [12]. Let us recall some terminology we shall need throughout this work (which can be found in [3, 4, 22, 24]). Assume that X is a vector space and \(\alpha \) is a cardinal number. Then a subset \(A\subset X\) is said to be:

lineable if there is an infinite dimensional vector space M such that \(M {\setminus } \{0\}\subset A\).

\(\alpha \)lineable if there exists a vector space M with \( \dim (M)=\alpha \) and \(M{\setminus } \{0\}\subset A\). In the case \(\dim (M)=\dim (X)\), the set A is called maximal lineable.
If, in addition, X is a topological vector space, then the subset A is said to be:

spaceable in X whenever there is a closed infinitedimensional vector subspace M of X such that \(M{\setminus } \{0\}\subset A\). In the case \(\dim (M)=\dim (X)\), the set A is called maximal spaceable.

\(\alpha \)latticeable if there exists a Riesz space M such that \(M\backslash \{0\}\subset A\) and M is an \(\alpha \)dimensional vector space. If, in addition, M is closed then A is said to be \(\alpha \)spaceable latticeable. Even more, it is said to be maximal whenever \(\dim (M)=\dim (X)\).

M is said to be \(\mu \)denselineable if \(M\cup \{0\}\) contains a dense vector space of dimension \(\mu \).
This paper is arranged as follows. Section 2 focuses on studying the lineability properties of certain subsets of continuous functions on bounded intervals. Within this class we consider, among several others: (i.) \(C^\infty (]0,1])\), i.e., the class of all elements of C([0, 1]) which are infinitely differentiable on ]0, 1], (ii.) \(\mathcal {D}_0\), i.e., the class of all functions \(h \in C([0,1])\) which are differentiable \(\lambda \)almost everywhere with derivative 0 but are not Lipschitz continuous, or (iii.) \(\mathcal {H}\), that is, the family of all functions \(h \in C([0,1])\) which are \(\alpha \)Hölder continuous for every \(\alpha \in ]0,1[\) but not Lipschitz continuous.
Next, if we fix an integer \(m\ge 0\), a real number \(1\le p<\infty \) and a vector space X, Sect. 3 studies the problem of finding lineable/latticeable subsets of the Sobolev Space \(W^{m,p}\left( ]0,1[\right) \). We shall provide a brief background on theory of Sobolev Spaces over one dimensional bounded intervals in order to have this section selfcontained. Section 4 considers the unbounded counterpart of Sect. 3 for Sobolev Spaces. The notation used throughout the paper shall be rather usual.
2 Lineability and spaceability in C([0, 1])
We consider the Banach space \((l^\infty ,\Vert \cdot \Vert _\infty )\) of all bounded sequences in \(\mathbb {R}\) as well as the Banach space \(({\text {Bd}}([0,1]),d_\infty )\) of all bounded functions, and the Banach space \((C([0,1],d_\infty )\) of all realvalued continuous functions on [0, 1], endowed with the uniform distance \(d_\infty \), respectively (we write \(d_\infty \) instead of \(\Vert \cdot \Vert _\infty \) to avoid the usage of one symbol for two different objects). In the sequel we will also view each of these Banach spaces as Banach algebras and lattices (with the usual coordinatewise/pointwise operations). The same holds for the closed subspace \(c_0\) of \(l^\infty \) containing all sequences in \(l^\infty \) converging to 0.
Throughout this section \(f:[0,1] \rightarrow [0,1]\) will denote a function for which there exists \(x_0 \in ]0,1[\) with \(f(x_0)=1\). The only exception is the proof of Theorem 2.10 in which f is \([1,1]\)valued. Writing \(\mathbf {s}=(s_1,s_2,\ldots ) \in l^\infty \) and \(I_n=]\frac{1}{2^n},\frac{1}{2^{n1}}[\) for every \(n \in \mathbb {N}\)), define the operator \(\Phi _f: l^\infty \rightarrow {\text {Bd}}([0,1])\) by
The subsequent lemma gathers the most important properties of \(\Phi _f\) and is straightforward to prove:
Lemma 2.1
\(\Phi _f\) is a linear isometric embedding of \((l^\infty ,\Vert \cdot \Vert _\infty )\) in \(({\text {Bd}}([0,1]),d_\infty )\) and at the same time an algebra and lattice isomorphism. Moreover, \( \Phi _f(l^\infty )\) is a closed subspace (algebra, lattice) of \(({\text {Bd}}([0,1]),d_\infty )\). The same holds for \(\Phi _f\) if \((l^\infty ,\Vert \cdot \Vert _\infty )\) is replaced by \((c_0,\Vert \cdot \Vert _\infty )\). Additionally, for \(\mathbf {s} \in c_0\) the function \(\Phi _f(\mathbf {s})\) is continuous at 0.
Proof
The fact that \(\Phi _f\) is a structurepreserving isometry on \((l^\infty ,\Vert \cdot \Vert _\infty )\) (and hence also on \((c_0,\Vert \cdot \Vert _\infty )\)) is straightforward to verify (the fact that \(f(x_0) = 1\) is crucial to guarantee this since, otherwise, we would not have an isometry, but a contraction). Considering that \((l^\infty ,\Vert \cdot \Vert _\infty )\) is complete and that \(\Phi _f\) is an isometry it follows that \(\Phi _f(l^\infty )\) is complete and consequently closed in \(({\text {Bd}}([0,1]),d_\infty )\). The same reasoning applies to \((c_0,\Vert \cdot \Vert _\infty )\).
Finally, continuity of \(\Phi _f(\mathbf {s})\) at 0 for \(\mathbf {s} \in c_0\) is a straightforward consequence of the fact that \(\vert \Phi _f(\mathbf {s})(x) \mathbf {1}_{I_n}(x) \vert \le \vert s_n \vert \) holds for every \(n \in \mathbb {N}\) and for every \(x \in [0,1]\). \(\square \)
Selecting f adequately yields various results on spaceability/lineability/latticeability of seemingly ‘small’ subfamilies of \(({\text {Bd}}([0,1]),d_\infty )\) and \((C([0,1]),d_\infty )\). In order to simplify notation we will let \(\mathcal {N}\) denote the class of nonmeasurable, bounded functions on [0, 1]. Recall that the notation \(C^\infty (]0,1])\), \(\mathcal {D}_0\), and \(\mathcal {H}\) was already presented in the previous section.
Theorem 2.2
The set \(C^\infty (]0,1])\) is maximal spaceable and latticeable in \((C([0,1]),d_\infty )\).
Proof
Let \(g:\mathbb {R} \rightarrow [0,1]\) be defined by \(g(t)=e^{\frac{1}{(t(1t))^2}}\) and set \(f=\frac{1}{g(\frac{1}{2})} \, g\). Then f maps [0, 1] into [0, 1] and fulfills \(f(0)=f(1)=0\) as well as \(f^{(n)}(0)=f^{(n)}(1)=0\) for every \(n \in \mathbb {N}\) where \(f^{(n)}\) denotes the derivative of order \(n \in \mathbb {N}\). For every \(\mathbf {s}\in l^\infty \) it therefore follows that \(\Phi _f(\mathbf {s})\) is infinitely differentiable on (0, 1]. According to Lemma 2.1\(\Phi _f(\mathbf {s})\) is continuous at 0 for every \(\mathbf {s} \in c_0\), which, again using Lemma 2.1, altogether yields that \(\Phi _f(c_0)\) is a closed subspace and sublattice of \((C([0,1]),d_\infty )\). This completes the proof. \(\square \)
The next theorem has already been established and proved in [20]—our approach using \(\Phi _f\), however, allows for an alternative very short and simple proof.
Theorem 2.3
The set \(\mathcal {D}_0\) is maximal spaceable and latticeable in \((C([0,1]),d_\infty )\).
Proof
Let \(F:[0,1] \rightarrow [0,1]\) denote a nondecreasing continuous function with \(F(0)=0,F(1)=1\) fulfilling \(F'(x)=0\) for \(\lambda \)almost every \(x \in [0,1]\). We could, for instance, choose F as the famous Cantor function (a.k.a. evil’s staircase, e.g., [14]) or Minkowski’s questions mark function (see, e.g., [17, 20] and the references therein) or work with fractal interpolation (see [26]). Obviously each such F corresponds to a probability measure which is singular w.r.t. \(\lambda \), and is not Lipschitz continuous. Choose such a function F and define the function \(f: [0,1] \rightarrow [0,1]\) by
Then \(f \in C([0,1])\), f is differentiable \(\lambda \)everywhere with derivative 0, and f is not Lipschitz continuous. As direct consequence of the construction of \(\Phi _f\) the same is true for \(\Phi _f(\mathbf {s})\) if we consider \(\mathbf {s} \in c_0\). Figure 1 depicts two examples of singular functions F, as well as the corresponding functions f and \(\Phi _f(\mathbf {s})\) for some \(\mathbf {s} \in c_0\). According to Lemma 2.1\(\Phi _f(c_0)\) is a closed subspace and sublattice of \((C([0,1]),d_\infty )\), so the proof is complete. \(\square \)
Theorem 2.4
The set \(\mathcal {H}\) is maximal spaceable and latticeable in \((C([0,1]),d_\infty )\).
Proof
Letting \(T:[0,1] \rightarrow [0,1]\) denote Takagi function (see [25]), defined by
whereby \(d(y,\mathbb {Z}):=\min \{d(y,z):z \in \mathbb {Z}\}\) and setting \(f:=\frac{3}{2}\,T\) yields a function \(f:[0,1] \rightarrow [0,1]\) with \(\max _{x \in [0,1]}f(x)=1\), which is \(\alpha \)Hölder continuous for every \(\alpha \in (0,1)\) but not Lipschitz continuous (again see [25]). Considering \(\Phi _f\) and proceeding as in the last two proofs yields the desired result. \(\square \)
Theorem 2.5
The set \(\mathcal {N}\) is maximal spaceable and latticeable in \(({\text {Bd}}([0,1]),d_\infty )\).
Proof
Letting \(N \subseteq ]0,1[\) denote a nonmeasurable set, setting \(f=\mathbf {1}_N\) as well as \(\Phi _f\) and proceeding as before directly yields the desired result. \(\square \)
Considering the following slight modification of \(\Phi _f\) allows for an alternative simple proof of the fact that the set of all functions \(f \in C([0,1])\) whose graph has Hausdorff and BoxCounting dimension equal to some fixed \(s \in ]1,2]\) is \(\mathfrak {c}\)lineable and latticeable in C([0, 1]) (see [6]).
Using the same notation as for \(\Phi _f\) define the operator \(\Psi _f: l^\infty \rightarrow {\text {Bd}}([0,1])\) by
Then \(\Psi _f\) is welldefined, obviously linear, injective, and Lipschitz continuous with Lipschitz constant \(L=1\) but no isometry. Before focusing on the aforementioned result we prove the following simple lemma which will be used afterwards.
Lemma 2.6
Suppose that \(f: [0,1] \rightarrow [0,1]\) fulfills that its graph \(\Gamma (f)\) has Hausdorff and Box–Counting dimension equal to \(\alpha \in (1,2]\) and let \(\Psi _f\) be defined according to Eq. (2.3). Then for every \(\mathbf {s} \in l^\infty \) the same is true for \(\Psi _f(\mathbf {s})\), i.e.,
holds.
Proof
Fix \(\mathbf {s} \ne \mathbf {0}\) and set \(M:= \max \{\Vert \mathbf {s} \Vert _\infty ,1\}\). Some biLipschitz argument in combination with countable stability of the Hausdorff dimension implies \(dim_H(\Gamma (\Psi _f(\mathbf {s}))) = \alpha \). It therefore suffices to show that the upper BoxCounting dimension^{Footnote 1} (see [16]) fulfills \(\overline{dim_B}(\Gamma (\Psi _f(\mathbf {s}))) \le \alpha \) which can be done as follows: According to [16] in the calculation of the boxcounting dimension it suffices to work with \(\delta _k=\frac{M}{2^k}\) meshes and \(k \in \mathbb {N}\). Since the set
can be covered by one square of side length \(\delta _k\), the minimum number \(N_{\delta _k} \left( \Gamma (\Psi _f(\mathbf {s})) \right) \) of squares of side length \(\delta _k\) needed to cover \(\Gamma (\Psi _f(\mathbf {s}))\) fulfills
which yields
\(\square \)
Lemma 2.6 directly yields the following result already proved in a different manner in [6]. Notice that (contrary to the results on the previous pages) we do not get spaceability since \(\Psi _f\) is not an isometry and we can not simply conclude that the subspace \(\Psi _f(l^\infty )\) is closed.
Theorem 2.7
The family of all functions \(f \in C([0,1])\) whose graph has Hausdorff and BoxCounting dimension equal to some fixed \(s \in ]1,2]\) is \(\mathfrak {c}\) lineable and latticeable in \((C([0,1]),d_\infty )\).
Focusing exclusively on the Hausdorff dimension, working with \(\Phi _f: c_0 \rightarrow {\text {Bd}}([0,1])\) for some continuous \(f:[0,1] \rightarrow [0,1]\) whose graph \(\Gamma (f)\) fulfills \(dim_H(\Gamma (f))=s \in [1,2]\), and again using some biLipschitz argument together with countable stability of the Hausdorff dimension even yields spaceability:
Theorem 2.8
The family of all functions \(f \in C([0,1])\) whose graph has Hausdorff dimension equal to some fixed \(s \in ]1,2]\) is spaceable and latticeable in \((C([0,1]),d_\infty )\).
Considering yet another small modification of \(\Phi _f\) allows for quick alternative proofs for some of the results going back to [18, 19]. In fact, setting
analogously to \(\Psi _f\) the new operator \(\hat{\Psi }_f\) is welldefined, linear, injective, and Lipschitz continuous with Lipschitz constant \(L=1\) (but not an isometry). Using \(\hat{\Psi }_f\) we can show the subsequent result (Theorem 2.3 combined with Corollary 2.1 in [18])—thereby \(\mathcal {D}_{dis}\) denotes the set of all functions \(h \in C([0,1])\) which are differentiable on [0, 1] (at 0 and 1 we consider the onesided derivates) with a derivative that is discontinuous at every point of a set with positive \(\lambda \)measure, and \(\mathcal {D}_{\lnot R}\) the family of all functions \(h \in C([0,1])\) which are differentiable on [0, 1] with a derivative that is bounded but not Riemann integrable.
Theorem 2.9
\(\mathcal {D}_{dis}\) and \(\mathcal {D}_{\lnot R}\) are \(\mathfrak {c}\)lineable and latticeable in \((C([0,1]),d_\infty )\).
Proof
(i) The assertion concerning \(\mathcal {D}_{dis}\) can be proved as follows: Let \(f_h\) denote one of the functions constructed in the proof of Theorem 2.3. in [18] and proceed as follows. Defining \(g:[0,1] \rightarrow [0,\infty )\) by
and setting \(f(x):= \frac{g(x)}{\max \{g(z): z \in [0,1]\}}\) yields a nonnegative, continuous function \(f: [0,1] \rightarrow [0,1]\) with \(f(0)=f(1)=0\), which attains its maximum 1 in (0, 1), which is differentiable on [0, 1] and fulfills that its derivative \(f'\) is discontinuous on a fat Cantor set C. As a direct result, the function \(\hat{\Psi }_f(\mathbf {s})\) is obviously differentiable on (0, 1]. Considering that the very definition of \(\hat{\Psi }_f\) implies that \(\hat{\Psi }_f(\mathbf {s})\) is also differentiable with (righthand) derivative 0 at 0 it follows that \(\hat{\Psi }_f(\mathbf {s})\) is differentiable on [0, 1]. Moreover, if \(s_n \ne 0\) for some \(n \in \mathbb {N}\), then \((\hat{\Psi }_f(\mathbf {s}))'\) is discontinuous on some fat Cantor set contained in \(I_n\), implying \(\hat{\Psi }_f(\mathbf {s}) \in \mathcal {D}_{dis}\).
(ii) The assertion concerning \(\mathcal {D}_{\lnot R}\) follows in the same fashion. \(\square \)
We now turn to the family \(\mathcal {M}_s \subseteq C([0,1])\) of all functions f fulfilling that the sets \(\underline{U}_h, \overline{U}_h\), defined by
both have Hausdorff dimension \(s \in ]0,1[\) and, again using \(\Phi _f\) for some appropriately chosen f, show that \(\mathcal {M}_s\) is spaceable in C([0, 1]). Notice that in this context we do not obtain latticeability since the range of the constructed function f is [0, 1].
Theorem 2.10
\(\mathcal {M}_s\) is spaceable in \((C([0,1]),d_\infty )\) for every \(s \in [0,1]\).
Proof
Since the assertion is trivial for \(s \in \{0,1\}\) it suffices to consider \(s \in (0,1)\). Fix \(\beta \in ]0,\frac{1}{2}[\) and define the contractions \(w_1,w_2: [0,1] \rightarrow [0,1]\) by \(w_1(x)= \beta x\) and \(w_2(x)= \beta x + 1 \beta \). Considering the Iterated Function System (IFS, for short) \(\{w_1,w_2\}\) and using the standard properties of IFSs (see [16] and [21]) it follows that there exists a unique nonempty compact subset \(C_\beta ^*\) of [0, 1] fulfilling
To simplify notation we will write \(\mathcal {W}(K) = w_1(K) \cup w_2(K)\) for every nonempty compact subset K of [0, 1] and refer to \(\mathcal {W}\) as Hutchinson operator induced by the IFS. Again following [16] and [21] and considering that \(w_1,w_2\) are similarities the set \(C_\beta ^*\) is selfsimilar and its Hausdorff dimension \(\text {dim}_H(C_\beta ^*)\) is the unique solution s of the equation \(2\beta ^s=1\), i.e., \(\text {dim}_H(C_\beta ^*)=\frac{\log (2)}{\log (\beta )} \in (0,1)\).
As next step we construct a continuous function \(g:[0,1] \rightarrow [0,1]\) fulfilling \(\overline{U}_g=C_\beta ^*\) in several steps. Set \(g_1(x)=1\) for every \(x \in [0,1]\) and define the function \(g_2: [0,1] \rightarrow [0,1]\) by
The basic idea for the construction of \(g_2\) is to replace \(g_1\) on the (open) interval \([0,1]{\setminus } \mathcal {W}([0,1])\) by a reflected tent map. Modifying \(g_2\) on each of the intervals constituting \([0,1] {\setminus } \mathcal {W}^2([0,1])\) in the same manner yields the function \(g_3\). Proceeding analogously yields a sequence \((g_n)_{n \in \mathbb {N}}\) of continuous functions that converges uniformly to a continuous function g, which obviously fulfills \(\overline{U}_g=C_\beta ^*\). The first and the second panel in the first row of Fig. 2 depict \(g_2\) and \(g_7\). Shrinking [0, 1] to \([\beta ,1\beta ]\) and extending linearly on both sides to 0 yields the continuous function \(g^*\) fulfilling \(g^*(0)=0=g^*(1)\) (third upper panel in Fig. 2). Finally defining \(f:[0,1] \rightarrow [1,1]\) by (see fourth panel in the first row of Fig. 2)
it follows that f is continuous, fulfills \(f(0)=0=f(1)\), and attains its maximum 1 and its minimum \(1\) both in sets of Hausdorff dimension \(\frac{\log (2)}{\log (\beta )}\). Considering the induced operator \(\Phi _f: c_0 \rightarrow {\text {Bd}}([0,1])\) it follows that \(\Phi _f(\mathbf {s}) \in C([0,1])\) for every \(\mathbf {s} \in c_0\) and the assertion of the theorem follows for \(s=\frac{\log (2)}{\log (\beta )}\). Since \(\{\frac{\log (2)}{\log (\beta )}: \beta \in ]0,\frac{1}{2}[\}=]0,1[\) this completes the proof. The lower panel in Fig. 2 depicts \(\Phi _f(\mathbf {s})\) for \(\beta =\frac{3}{8}\) and \(\mathbf {s}=(s_1,s_2,s_3,\ldots )\) with \(s_i=\frac{1}{i} (1)^{i+1}\) \(\square \)
Remark 2.11
An alternative simple way for constructing the (Lipschitz) continuous function \(g^*\) used in the proof of Theorem 2.10 would be as follows: Letting \(\mathcal {F}\) denote the family of functions in C([0, 1]) fulfilling \(f(0)=f(1)=1\) it follows that \(\mathcal {F}\) is closed in \((C([0,1]),d_\infty )\). Defining the operator \(T_\beta : \mathcal {F} \rightarrow \mathcal {F}\) by
it is straightforward to verify that \(T_\beta \) is welldefined and a contraction on \((\mathcal {F},d_\infty )\), so Banach’s Fixed Point Theorem implies the existence of a unique, globally attractive fixed point, which is easily verified to coincide with \(g^*\).
We conclude this section by carrying over the main results from C([0, 1]) established so far to
and start with the \(L^p\)version \(\Phi _f^p\) of the operator \(\Phi _f\). For \(p \in [1,\infty )\), \(f \in L^p([0,1])\) and \(\mathbf {s} \in l^p\) define \(\Phi _f^p: l^p \rightarrow L^p([0,1])\) by
where the righthand side is to be interpreted as equivalence class in \(L^p([0,1])\). In the sequel we will write \(\Vert \cdot \Vert _p\) both for the norm on \(l^p\) and the norm on \(L^p([0,1])\) since no confusion will arise. Considering that for every \(f \in L^p([0,1])\) with \(\Vert f \Vert _p=1\) we have
it follows that for each such f the operator \(\Phi _f^p: l^p \rightarrow L^p([0,1])\) is a linear isometry and a lattice isomorphism. As a direct consequence, \(\Phi _f^p(l^p)\) is a closed subspace of \(L^p([0,1])\). Notice, however, that although obviously \(l^p \subseteq c_0\) holds, in general \(\Phi _f^p(\mathbf {s})\) is not necessarily continuous at 0 (and the same holds true for each representative of \(\Phi _f^p(\mathbf {s})\)). Working with the operator \(\Phi _f^p\) and preceding as before yields the following complementing results on lineability/spaceability of subsets of \(L^p([0,1])\):
Theorem 2.12
The following assertions hold for every fixed \(p \in [1,\infty )\):

1.
The set of equivalence classes in \(L^p([0,1])\) which contain some representative contained in \(C^\infty (]0,1])\) is spaceable and latticeable in \(L^p([0,1])\).

2.
The set of equivalence classes in \(L^p([0,1])\) which contain some representative that is differentiable \(\lambda \)almost everywhere with derivative 0 but not Lipschitz continuous is spaceable and latticeable in \(L^p([0,1])\).

3.
The set of equivalence classes in \(L^p([0,1])\) which contain some representative that is \(\alpha \)Hölder continuous for every \(\alpha \in ]0,1[\) but not Lipschitz continuous is spaceable and latticeable in \(L^p([0,1])\).

4.
The set of equivalence classes in \(L^p([0,1])\) which contain some representative whose graph has Hausdorff and Box–Counting dimension equal to some \(s \in ]1,2]\) is \(\mathfrak {c}\)lineable and latticeable in \(L^p([0,1])\).

5.
The set of equivalence classes in \(L^p([0,1])\) which contain some representative whose graph has Hausdorff dimension equal to some \(s \in ]1,2]\) is spaceable and latticeable in \(L^p([0,1])\).
3 Lineability, spaceability and latticeability in Sobolev spaces over bounded intervals
In this section we consider the problem of finding lineable/latticeable subsets of Sobolev Spaces over onedimensional bounded intervals. Thus, fixed an integer \(m\ge 0\) and a real number \(1\le p<\infty \) we consider, as vector space X, the Sobolev Space \(W^{m,p}\left( ]0,1[\right) \). Before recalling basic facts about Sobolev spaces over intervals we start with some preparations which will be used subsequently.
Let \((t_n)_{n \in \mathbb {N}}\) denote the Thue–Morse sequence (also known as Prouhet–Thue–Morse sequence) defined to be zero if the sum of the digits in the binary expansion of n is even and \(t_n=1\) otherwise. Let us recall that \(t_n\) is a binary digits sequence for which the sequence \(d_n = t_0t_1\cdots t_{2^{n}1}\) satisfies that \(d_{n+1}\) is the concatenation of \( d_n\) and is Boolean complementary, i.e. \(d_0 = 0, d_1= 01, d_2 = 0110, d_3 = 01101001, d_4 = 0110100110010110, \ldots \)
Since binary sequences are identified with \(\mathbb {Z}_{2}\) numbers, the number \(t = \sum _{n=0}^{\infty }t_{n}2^{n}\) associated to the sequence \( t_{n} \) is the unique fixed point of the contraction
For each fixed integer \(m\ge 1\), we will use the following functions in \(\mathbb {R}^{[0,1]}\):
In addition, for each \(j\in \{1, \ldots , m\}\), we recursively define the function \(s_{m,j}\in \mathbb {R}^{[0,1]}\) by
In the following lemma we collect some elementary properties of the functions \(s_{m,j}\).
Lemma 3.1
For every integer \(m\ge 1\) the function \(s_{m,m}\) is nonnegative and

(a)
\(s_{m, j}\) is bounded for every \(j \in \left\{ 0, 1,\ldots ,m\right\} \),

(b)
\(s_{m,j}(0) = s_{m,j}(1)=0\) for every \(j\in \left\{ 1,\ldots ,m\right\} \).
We now recall some basic definitions and main properties of Sobolev spaces in one dimension, see [7] for an extended study using weak derivatives (for a distributional point of view see also [23]). We recall that, given \(m\in \mathbb {N}\) and \(1\le p \le \infty \) the Sobolev space \(W^{m,p}\left( ]a,b[\right) \) can be defined as follows
where \(C_{c}^{\infty }(]a,b[)\) denotes the space of compactly supported, infinitely differentiable functions on ]a, b[.
Also, the function \(g_j\) involved in the previous definition is the well known weak derivative of jorder of the function u and is denoted as usual by \(g_j\equiv u^{j)}\).
The standard norm in the Sobolev space \(W^{m,p}\left( ]a,b[\right) \) is given by
where \(\Vert \cdot \Vert _p\) denotes the usual \(L^p\)norm. The space \(W^{m,p}\left( ]a,b[\right) \) endowed with the norm \(\Vert \cdot \Vert _{m,p}\) satisfies the following basic properties:
Lemma 3.2
Let \(m\in \mathbb {N}\), \(1\le p\le \infty \) and \(]a,b[\subset \mathbb {R}\). Then:

1.
\(W^{m,p}\left( ]a,b[\right) \) is a Banach space.

2.
\(W^{m,p}\left( ]a,b[\right) \) is reflexive for \(1< p < \infty \).

3.
\(W^{m,p}\left( ]a,b[\right) \) is separable for \(1 \le p < \infty \).

4.
For every \(1\le p<\infty \), \(C_c^{\infty }(\mathbb {R})\) is dense \(W^{m,p}\left( ]a,b[\right) \) with respect to the norm \(\Vert \cdot \Vert _{m,p}\).
In the particular case of ]a, b[ being bounded we have, in addition, that:

5.
For every \(1\le p<\infty \), the polynomials are dense in \(W^{m,p}\left( ]a,b[\right) \).^{Footnote 2}

6.
\(W^{m,p}\left( ]a,b[\right) \) is continuously embedded in \(C^{m1}([a,b])\). Moreover the embedding is compact for \(1<p \le \infty \).
The clousure of \(C_c^\infty (]a,b[)\) in the space \(W^{m,p}\left( ]a,b[\right) \), for \(1\le p<\infty \) is denoted by \(W_0^{m,p}\left( ]a,b[\right) \) and satisfies the following properties:
Lemma 3.3
Let \(m\in \mathbb {N}\), \(1\le p<\infty \) and \(]a,b[\subset \mathbb {R}\). Then:

1.
\(W_0^{m,p}\left( ]a,b[\right) \) is a separable Banach space.

2.
\(W_0^{m,p}\left( ]a,b[\right) \) is reflexive for \(1 < p \).

3.
\(u\in W^{m,p}\left( ]a,b[\right) \cap C^{m1}\left( \overline{]a,b[}\right) \) belongs to \(W_0^{m,p}\left( ]a,b[\right) \) if and only if \(u=u'=\cdots =u^{m1)}=0\) on the boundary of ]a, b[.

4.
When ]a, b[ is bounded, \(\Vert u^{m)}\Vert _p\) for \(u\in W_0^{m,p}\left( ]a,b[\right) \) is a norm equivalent to \(\Vert u\Vert _{m,p}\).
The main result in this section is the following.
Theorem 3.4
For every \(1\le p<+\infty \) the set \(W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \) is spaceable latticeable.
Proof
Fixed m and p, we consider the function \(f_{n}\in \mathbb {R}^{[0,1]}\) defined by
We observe that \(f_{n}\) has derivatives up to order m and \(f_{n}^{j)}\) is continuous for every \(j\in \{0, 1, \ldots , m1\}\) and \(f_{n}^{j)}\left( \frac{1}{2^{n}}\right) =f_{n}^{j)}\left( \frac{1}{2^{n1}}\right) =0\). Therefore, for every \(j=1,2,\ldots ,m\)
holds for all sufficiently regular \(\varphi \), implying
Now we consider, for every \(d\in \mathbb {N}\), an infinite subset \(A_d\subset \mathbb {N}\) such that \(\mathbb {N}=\bigcup _{d\in \mathbb {N}} A_d\) and any two of these subsets are disjoint. Let us denote by d(n) the element in the position n of \(A_d\) with the usual order. Moreover, we define the function \(\gamma _{d}\in \mathbb {R}^{[0,1]}\) by
where \(a_{n}:=\left( n\ln ^{2}(n+1)\right) ^{1/p}\).
We claim that \(\gamma _{d}\in W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \). In order to prove the claim we first show that \(\gamma _{d}\) admits weak derivatives up to order m. Indeed, given a function \(\varphi \) regular enough with \({{\,\mathrm{supp}\,}}\varphi \cap ] \frac{1}{2^{d(n)}},\frac{1}{2^{d(n)1}}[ \ne \emptyset \), we have, using the properties of \(f_{d(n)}\), that
In particular, given \(\varphi \in C_{c}^{\infty }(]0,1[)\) we deduce that
Since \(\gamma _{d}\) and its derivatives up to order \(m1\) are bounded functions we have that \(\gamma _{d}^{j)} \in L^{p}(]0,1[)\) for every \(j\in \{0, 1, \ldots , m1\}\). With respect to \(\gamma _{d}^{m)}\) we know that
In particular \(\gamma _{d}\in W^{m,p}\left( ]0,1[\right) \) since
However, \(\gamma _{d}^{m)}\) does not belong to \(L^{q}(]0,1[)\) when \(q>p\) since
This proves the claim, i.e. \(\gamma _{d}\in W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \).
Let us write \(h_{d}:=\gamma _{d}/\left\ \gamma _{d}\right\ _{m,p}\) and consider the set \(H\subset W^{m,p}\left( ]0,1[\right) \) given by
By construction, the coefficients \(c_d\) associated to any function \(h\in H\) are uniquely determined and H is a nontrivial vector space. We prove now that \(H{\setminus }\{0\} \subset W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \), H is closed in \(W^{m,p}\left( ]0,1[\right) \) and that it is a lattice.
In order to prove that H is closed we assume that \(g_r\in H\) and that \( \lim _{r\rightarrow \infty } \Vert g_r  g\Vert _{m,p} =0\) for some \(g\in W^{m,p}\left( ]0,1[\right) \). We may assume, if necessary, that \(g_r\) and g are continuous according to the Sobolev embedding (see (6), Lemma 3.2). In particular, since \( \left\ g_{r}g\right\ _{p} \le \Vert g_rg\Vert _{m,p}\), we have that
For each \(d\in \mathbb {N}\) there exists a unique coefficient \(c_{d_r}\in \mathbb {R}\) such that, given \(n\in \mathbb {N}\) we have that
Therefore it follows that
Now we claim that, fixed d, the sequence \(\left\{ c_{d_{r}}\right\} \) is bounded. Otherwise we may assume  up to a subsequence  that \(c_{d_{r}}\) is increasing and unbounded (observe that we may replace g by \(g\)). Using the definition of \(f_{d(n)}\) we may choose \(\omega > 0\) such that \(S_\omega = f_{d(n)}^{1}([\omega ,+\infty [)\) is a subset of \(] 2^{d(n)},2^{d(n)+1}[\) with positive measure. Moreover, g is bounded and \(f_{d(n)}(x)a_{n}\ge 0\) in the interval \(] 2^{d(n)},2^{d(n)+1} [\) and, as a particular case, in \(S_\omega \). Thus, given \(M>0\) there exists \(r_{M}>0 \) such that \(\left c_{d_{r}}\frac{f_{d(n)}(x)}{ \left\ \gamma _{d}\right\ _{m,p}}a_{n}g(x)\right ^{p}>M\) for every \(x \in S_\omega \) and every \(r>r_{M}\). This implies, using that the measure of \(S_\omega \) is positive, that
This is a contradiction and this completes the proof that \(\left\{ c_{d_{r}}\right\} \) is bounded. In addition, we may assume that  up to a subsequence  \(\left\{ c_{d_{r}}\right\} \) converges to some \(c_{d}\in \mathbb {R}\). In particular
This implies that the restriction of the function g to the interval \(] 2^{d(n)},2^{d(n)+1}[ \) is equal to \(c_{d}\frac{f_{d(n)}(x)}{ \left\ \gamma _{d}\right\ _{m,p}}a_{n}\) for every \(n\in \mathbb {N}\) (observe that \(c_{d_r}\) and \(c_d\) do not depend on n). Therefore g restricted to the set
is equal to \(c_{d}h_{d}\) or equivalently \(g=\sum _{d=1}^{\infty }c_{d}h_{d}\) and we have proved that H is closed.
Observe also that, by construction, given \(h\in H{\setminus }\{0\}\) then \( h=\sum _{d=1}^{\infty }c_{d}h_{d}\) with some \(c_{d}\ne 0\), thus, if \(h\in W^{m,q}\left( ]0,1[\right) \) for some \(q>p\) then the corresponding function \( h_{d}\) belongs to \(W^{m,q}\left( ]0,1[\right) \) which is a contradiction, i.e. \(H{\setminus }\{0\} \subset W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \).
Moreover H is infinite dimensional, actually we may define the injective map \(T : l_{\infty } \rightarrow H\) given by \(T\left( c\right) = \sum _{d=1}^{\infty }c_{d} h_{d}\) for every \(c=\{c_d\}\in l^\infty \). We observe that
which implies that T is well defined and even more, that \(l^\infty \) is continuously embedded in H.
In particular, H contain \(\mathfrak {c}\) independent vectors.
We finally show that H is a lattice. Indeed, given \(z,g\in H\) with \( z(x)=\sum _{d=1}^{\infty }c_{d}h_{d}(x)\) and \(g(x)=\sum _{d=1}^{\infty }g_{d}h_{d}(x)\) then \(z\vee g\) is given by \(\left( z\vee g\right) (x)=\sum _{d=1}^{\infty }\left( c_{d}\vee g_{d}\right) h_{d}(x)\). Observe that, for \(\varphi \) regular enough we have
In particular,
Moreover, since
we have that \(z\vee g\in W^{m,p}\left( ]0,1[\right) \) and \(\Vert z\vee g\Vert _{m,p}^{p}\le \Vert z\Vert _{m,p}^{p}+\Vert g\Vert _{m, p}^{p}\). Therefore \(z\vee g\in H\).
Finally the maximality is deduced from the fact that \(W^{m,p}\left( ]0,1[\right) \) is continuously embedded in C(]0, 1[). Thus the space \( W^{m,p}\left( ]0,1[\right) \) has cardinality \(\mathfrak {c}\). \(\square \)
Remark 3.5
Observe that, since the functions \(h_d\) in the proof above vanish at the boundary of ]0, 1[, from Lemma 3.3 it is deduced that in fact we have proved that, for every \(1\le p<+\infty \) the set
is maximal spaceable latticeable.
We conclude this section proving maximal denselineability by using the sufficient condition in [2] (see Theorem 3.7 bellow). Let us also recall the following definition from that paper.
Definition 3.6
Let A, B be subsets of a vector space X. We say that A is stronger than B if \(A+B\subseteq A\).
Theorem 3.7
[2] Let X be a separable Banach space, and consider two subsets A, B of X such that A is lineable and B denselineable. If A is stronger than B, then A is denselineable.
Now we can prove the maximal denselineability of \(W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \).
Corollary 3.8
For every \(1\le p<+\infty \) the set \(W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \) is maximal denselineable.
Proof
The proof follows immediately from Theorem 3.7 with \(X=W^{m,p}\left( ]0,1[ \right) \), the set A is given by \(W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \) which, according to Theorem 3.4, is in particular lineable, and finally the set B is the vector space of polynomials in ]0, 1[ which, by Lemma 3.2, is dense in \(W^{m,p}\left( ]0,1[ \right) \). Thus we only have to prove that A is stronger than B. i.e., \(A+B\subseteq A\). In order to prove that we observe that \(A+B\subseteq W^{m,p}\left( ]0,1[\right) \) and given \(f\in A\) and \(p_0\in B\) we have that \((f+p_0)\not \in \bigcup _{q>p}W^{m,q} \left( ]0,1[\right) \). Otherwise, for some \(q_0>p\), \((f+p_0) \in W^{m,q_0} \left( ]0,1[\right) \) and, since \(p_0\in W^{m,q_0} \left( ]0,1[\right) \), we have that \(f=(f+p_0)p_0 \in W^{m,q_0} \left( ]0,1[\right) \) which contradicts that \(f\in A\). \(\square \)
4 Lineability, spaceability and latticeability in Sobolev spaces over unbounded intervals
In this section we focus on the problem of finding latticeable subsets of Sobolev Spaces over one dimensional unbounded intervals, namely, the Sobolev Space \(W^{m,p}\left( ]1, +\infty [\right) \).
The analogues to the functions \(s_{m,j}\) are given by \(k_{m,j} \in \mathbb {R}^{ \left[ 0,2^{m}\right] }\) with
and, for \(j\in \{1, \ldots , m\}\),
The properties of functions \(k_{m,j}\) are analogous to those of \(s_{m,j}\) and are collected in the following lemma.
Lemma 4.1
For every integer \(m\ge 1\) the function \(k_{m,m}\) is nonnegative and

(a)
\(k_{m, j}\) is bounded for every \(j \in \left\{ 0, 1,\ldots ,m\right\} \),

(b)
\(k_{m,j}(0) = k_{m,j}(2^m)=0\) for every \(j\in \left\{ 1,\ldots ,m\right\} \).
The main result about latticeability on Sobolev Spaces over unbounded intervals is the following.
Theorem 4.2
For every \(1< p<+\infty \) the set \(W^{m,p}\left( ]1,+\infty [\right) \backslash \bigcup _{q<p}W^{m,q}\left( ]1,+\infty [\right) \) is spaceable latticeable.
Proof
Let us consider, for every \(e\in \mathbb {N}\), an infinite subset \( B_{e}\subset \mathbb {N}\) such that \(\mathbb {N}=\bigcup _{e\in \mathbb {N} }B_{e} \) and any two of these subsets are disjoint. Let us denote by e(n) the element in the position n of \(B_{e}\) with the usual order. The proof follows as in Theorem 3.4 replacing functions \(\gamma _{d}\) by
where, as before, \(a_{n}:=\left( n\ln ^{2}(n+1)\right) ^{1/p}\).
Write \(\eta _{e}:=\rho _{e}/\left\ \rho _{e}\right\ _{m,p}\) and consider now the set \(B \subset W^{m,p}\left( ]1, +\infty [\right) \) given by
Arguing as in the previous section the coefficients \(c_e\) associated to any function \(\rho \in B\) are uniquely determined and B is a vector space with \(B{\setminus }\{0\} \subset W^{m,p}\left( ]1, +\infty [\right) \backslash \bigcup _{q<p}W^{m,q}\left( ]1,+\infty [\right) \). Let us show that B is closed in \(W^{m,p}\left( ]1,+\infty [\right) \). Indeed, let us assume that \(h_r\in B\) and that \( \lim _{r\rightarrow \infty } \Vert h_r  h\Vert _{m,p} =0\) for some \(h\in W^{m,p}\left( ]1,+\infty [\right) \).
We may assume, if necessary, that \(h_r\) and h are continuous according to the Sobolev embedding. In particular, since \( \left\ h_{r}h\right\ _{p} \le \Vert h_rh\Vert _{m,p}\), we have that
For each \(e\in \mathbb {N}\) there exists a unique coefficient \(c_{e_r}\in \mathbb {R}\) such that, given \(n\in \mathbb {N}\) we have that
Therefore we have that, for every \(e,n\in \mathbb {N}\),
Now we claim that, fixed \(e\in \mathbb {N}\), the sequence \(\left\{ c_{e_{r}}\right\} \) is bounded. We will use that for every fixed \(n\in \mathbb {N}\) we have
Assume on the contrary that  up to a subsequence  \(c_{e_{r}}\) is increasing and unbounded (observe that we may replace h by \(h\)). Since \(k_{m,m}\) is nonnegative and nontrivial we may choose \(\omega > 0\) such that \(S_\omega = k_{m,m}^{1}([\omega ,+\infty [)\) is a positive measure subset of \(] 0,2^{m}[\). Moreover, since h is bounded on compact sets,
for some positive constant \(C_{n,e}\in \mathbb {R}\) not depending on r. Thus, given \(M>0\) there exists \(r_0\equiv r_0(M,e,n)>0 \) such that, for every \(t\in S_\omega \),
This implies, considering that the measure of \(S_\omega \) is positive, that
This is a contradiction and thus we have the claim proved, i.e. \(\left\{ c_{e_{r}}\right\} \) is bounded. In addition, we may assume that  up to a subsequence  \(\left\{ c_{e_{r}}\right\} \) converges to some \(c_{e}\in \mathbb {R}\). In particular
This implies that the restriction of the function h to the interval \(] 2^{e(n)1},2^{e(n)}[ \) is equal to
for every \(n\in \mathbb {N}\) (observe that \(c_{e_r}\) and \(c_e\) do not depend on n). Therefore h restricted to the set
is equal to \(c_{e}\eta _{e}\) or equivalently \(\displaystyle h=\sum \nolimits _{e=1}^{\infty }c_{e}\eta _{e}\) and we have proved that B is closed. \(\square \)
The next result is obtained by combining the results of Theorems 3.4 and 4.2.
Theorem 4.3
For every \(1<p<+\infty \) the set \(W^{m,p}\left( ]0,+\infty [\right) \backslash \bigcup _{q\ne p}W^{m,q}\left( ]0,+\infty [\right) \) is latticeable.
Remark 4.4
Arguing as was pointed out in Remark 3.5 the same result is also true for the set
Proof
Let us denote by \(\lambda _{s}\) with \(s\in [0,1]\) to a family of \( \mathfrak {c}\) linearly independent functions in the set \(W^{m,p}\left( ]0,1[\right) \backslash \bigcup _{q>p}W^{m,q}\left( ]0,1[\right) \) whose existence is guaranteed by Theorem 3.4. Moreover, we may assume that \( \lambda _{s}\in C([0,1])\) and \(\lambda _{s}(0)=\lambda _{s}(1)=0\). Similarly we denote by \(\upsilon _{s}\) with \(s\in [0,1]\) to a family of \( \mathfrak {c}\) linearly independent functions in the set \(W^{m,p}\left( ]1,+\infty [\right) \backslash \bigcup _{q<p}W^{m,q}\left( ]1,+\infty [\right) \) whose existence ensure Theorem 4.2. In this case we may assume that \(\upsilon _{s}\in C([1,+\infty [)\) and \(\upsilon _{s}(1)=0\).
Next we define the functions
which generate a vector space contained in \(W^{m,p}\left( ]0,+\infty [\right) \backslash \bigcup _{q\ne p}W^{m,q}\left( ]0,+\infty [\right) \). In particular, this set is maximal latticeable. \(\square \)
As a corollary we deduce the same result for Sobolev Spaces in the real line.
Corollary 4.5
For every \(1<p<+\infty \) the set \(W^{m,p}\left( \mathbb {R}\right) \backslash \bigcup _{q\ne p}W^{m,q}\left( \mathbb {R}\right) \) is latticeable.
Corollary 4.6
Assume that I is an unbounded open interval and \(1\le p<+\infty \). Then, the set
is maximal denselineable.
Proof
Assume without loss of generality that \(I=\mathbb {R}\). We use again Theorem 3.7 with \(X=W^{m,p}\left( \mathbb {R} \right) \), \(A=W^{m,p}\left( \mathbb {R} \right) \backslash \bigcup _{q \ne p}W^{m,q}\left( \mathbb {R} \right) \) which, according to Corollary 4.5, is in particular lineable, and \(B=C_c^\infty (\mathbb {R})\) which, by Lemma 3.2, is dense in \(W^{m,p}\left( \mathbb {R} \right) \). We now show that A is stronger than B. i.e., \(A+B\subseteq A\). First we observe that \(A+B\subseteq W^{m,p}\left( \mathbb {R} \right) \) and, given \(f\in A\) and \(g\in B\) we have that \((f+g)\not \in \bigcup _{q\ne p}W^{m,q} \left( \mathbb {R} \right) \). Otherwise, for some \(q_0\ne p\), \((f+g) \in W^{m,q_0} \left( \mathbb {R}\right) \) and, since \(g\in W^{m,q_0} \left( \mathbb {R}\right) \), it follows that \(f=(f+g)g \in W^{m,q_0} \left( \mathbb {R}\right) \) which contradicts that \(f\in A\). \(\square \)
We conclude this section by extending the obtained results to \(\mathbb {R}^N\). In fact, the results of the previous section can be easily extended to the case of Sobolev spaces \(W^{m,p}(I)\) where \(I=I_1\times \cdots \times I_N\) is a Ndimensional bounded cube of \(\mathbb {R}^N\). Indeed, since I is bounded we have \(W^{m,p}(I_i)\subset W^{m,p}(I)\) by considering extensions \(\tilde{w}(x_1,\ldots , x_N)= w(x_i)\) for each \(w\in W^{m,p}(I_i)\). In addition \(\Vert {\tilde{w}}\Vert _{m,p} \le C \Vert w\Vert _{m,p}\). Thus, since \(\displaystyle W^{m,p}(I_i){{\setminus }} \bigcup _{q>p}W^{m,q}(I_i)\) is spaceable latticeable and
we have that \(W^{m,p}(I){{\setminus }} \bigcup _{q>p}W^{m,q}(I)\) is spaceable latticeable.
Moreover, for any Ndimensional cube I, non necessarily bounded, we have that \(W_0^{m,p}(I_1)\times \cdots \times W_0^{m,p}(I_N)\subset W_0^{m,p}(I)\) by considering \(\tilde{w}(x_1,\ldots , x_N)= \prod _{i=1}^N w_i(x_i)\) for every \((w_1, \ldots , w_N)\in W_0^{m,p}(I_1)\times \cdots \times W_0^{m,p}(I_N)\). In addition, \(\Vert {\tilde{w}}\Vert _{m,p} \le C \prod _{i=1}^N \Vert w_i\Vert _{m,p}\). Thus, if \(I_1\) is bounded, using that
it follows that \(W_0^{m,p}(I){{\setminus }} \bigcup _{q>p}W_0^{m,q}(I)\) is spaceable latticeable.
Analogously, if \(I_1\) is unbounded we have that \(W_0^{m,p}(I){\setminus } \bigcup _{q\ne p}W_0^{m,q}(I)\) is spaceable latticeable for every \(1<p<\infty \). These results are also true for a general open subset \(\Omega \subset \mathbb {R}^N\) since for every Ndimensional cube \(I\subset \Omega \), \(W_0^{m,p}(I)\subset W_0^{m,p}(\Omega )\) by means of extending functions by zero.
Collecting everything we have proved the following result.
Theorem 4.7
Assume \(I=I_1\times \cdots \times I_N\) that for some real intervals, \(I_1, \ldots , I_N\). Then

1.
If I is bounded, \(W^{m,p}(I){{\setminus }} \bigcup _{q>p}W^{m,q}(I)\) is spaceable latticeable.

2.
If I is bounded in one direction, \(W_0^{m,p}(I){{\setminus }} \bigcup _{q>p}W_0^{m,q}(I)\) is spaceable latticeable.

3.
If I is unbounded in one direction, \(W_0^{m,p}(I){{\setminus }} \bigcup _{q\ne p}W_0^{m,q}(I)\) is spaceable latticeable.

4.
If \(\Omega \subset \mathbb {R}^N\) is open then \(W_0^{m,p}(\Omega ){{\setminus }} \bigcup _{q>p}W_0^{m,q}(\Omega )\) is spaceable latticeable.

5.
If \(\Omega \subset \mathbb {R}^N\) is open and it contains an unbounded cube I then
$$\begin{aligned} W_0^{m,p}(\Omega ){{\setminus }} \bigcup _{q\ne p}W_0^{m,q}(\Omega ) \end{aligned}$$is spaceable latticeable.
Notes
Given any nonempty subset M of \(\mathbb {R}^n\), we let \(N_\delta (M)\) denote the smallest number of sets, of diameter at most \(\delta \), needed to cover M. Then
$$\begin{aligned} \overline{dim_B} (M):=\limsup _{\delta \rightarrow 0} \frac{\log N_\delta (M)}{\log \delta }. \end{aligned}$$Item (5) in Lemma 3.2 follows straightforwardly by using item (4) and Bernstein’s proof of the Weierstrass theorem, where Bernstein’s polynomials approximate \(C^k([a,b])\) functions in \(C^k([a,b])\).
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Acknowledgements
José Carmona Tapia is partially supported by Ministerio de Ciencia, Innovación y Universidades (MCIU), Agencia Estatal de Investigación (AEI) and (FEDER) Fondo Europeo de Desarrollo Regional under Research Project PGC2018096422BI00, Junta de Andalucía, Consejería de Transformación Económica, Industria, Conocimiento y UniversidadesUnión Europea Grant P18FR667, FQM194 and CDTIME. Juan B. SeoaneSepúlveda gratefully acknowledges the support of Grant PGC2018097286BI00. Wolfgang Trutschnig gratefully acknowledges the support of the WISS 2025 project ‘IDAlab Salzburg’ (20204WISS/225/1972019 and 20102F1901166KZP).
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Carmona Tapia, J., FernándezSánchez, J., SeoaneSepúlveda, J.B. et al. Lineability, spaceability, and latticeability of subsets of C([0, 1]) and Sobolev spaces. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. AMat. 116, 113 (2022). https://doi.org/10.1007/s1339802201256y
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DOI: https://doi.org/10.1007/s1339802201256y