Lineability, spaceability, and latticeability of subsets of C([0, 1]) and Sobolev spaces

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).

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 infinite-dimensional vector subspace M of X such that M\{0} ⊂ A. In the case dim(M) = dim(X ), the set A is called maximal spaceable. • α-latticeable if there exists a Riesz space M such that M\{0} ⊂ A and M is an αdimensional vector space. If, in addition, M is closed then A is said to be α-spaceable latticeable. Even more, it is said to be maximal whenever dim(M) = dim(X ). Next, if we fix an integer m ≥ 0, a real number 1 ≤ p < ∞ and a vector space X , Sect. 3 studies the problem of finding lineable/latticeable subsets of the Sobolev Space W m, p (]0, 1[). We shall provide a brief background on theory of Sobolev Spaces over one dimensional bounded intervals in order to have this section self-contained. Section 4 considers the unbounded counterpart of Sect. 3 for Sobolev Spaces. The notation used throughout the paper shall be rather usual.

Lineability and spaceability in C([0, 1])
We consider the Banach space (l ∞ , · ∞ ) of all bounded sequences in R as well as the Banach space (Bd([0, 1]), d ∞ ) of all bounded functions, and the Banach space (C([0, 1], d ∞ ) of all real-valued continuous functions on [0, 1], endowed with the uniform distance d ∞ , respectively (we write d ∞ instead of · ∞ 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 coordinate-wise/pointwise operations). The same holds for the closed subspace c 0 of l ∞ containing all sequences in l ∞ converging to 0.
Throughout this section f : [0, 1] → [0, 1] will denote a function for which there exists x 0 ∈]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 s = (s 1 , s 2 , . . .) ∈ l ∞ and I n =] 1 2 n , 1 2 n−1 [ for every n ∈ N), define the operator f : l ∞ → Bd([0, 1]) by The subsequent lemma gathers the most important properties of f and is straightforward to prove: and at the same time an algebra-and lattice isomorphism.
Additionally, for s ∈ c 0 the function f (s) is continuous at 0.

Proof
The fact that f is a structure-preserving isometry on (l ∞ , · ∞ ) (and hence also on (c 0 , · ∞ )) 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 ∞ , · ∞ ) is complete and that f is an isometry it follows that f (l ∞ ) is complete and consequently closed in (Bd([0, 1]), d ∞ ). The same reasoning applies to (c 0 , · ∞ ).
Finally, continuity of f (s) at 0 for s ∈ c 0 is a straightforward consequence of the fact that | f (s)(x)1 I n (x)| ≤ |s n | holds for every n ∈ N and for every x ∈ [0, 1].
Selecting f adequately yields various results on spaceability/lineability/latticeability of seemingly 'small' subfamilies of (Bd . In order to simplify notation we will let N denote the class of non-measurable, bounded functions on [0, 1]. Recall that the notation C ∞ (]0, 1]), D 0 , and H was already presented in the previous section. 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 ∈ N where f (n) denotes the derivative of order n ∈ N. For every s ∈ l ∞ it therefore follows that f (s) is infinitely differentiable on (0, 1]. According to Lemma 2.1 f (s) is continuous at 0 for every s ∈ c 0 , which, again using Lemma 2.1, altogether yields that f (c 0 ) is a closed subspace and sublattice of (C([0, 1]), d ∞ ). This completes the proof.
The next theorem has already been established and proved in [20]-our approach using f , however, allows for an alternative very short and simple proof. 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]  Proof Letting T : [0, 1] → [0, 1] denote Takagi function (see [25]), defined by whereby d(y, Z) := min{d(y, z) : z ∈ Z} and setting f := 3 2 T yields a function f : 1] f (x) = 1, which is α-Hölder continuous for every α ∈ (0, 1) but not Lipschitz continuous (again see [25]). Considering f and proceeding as in the last two proofs yields the desired result. Proof Letting N ⊆]0, 1[ denote a non-measurable set, setting f = 1 N as well as f and proceeding as before directly yields the desired result.
Considering the following slight modification of f allows for an alternative simple proof of the fact that the set of all functions f ∈ C([0, 1]) whose graph has Hausdorff-and Box-Counting dimension equal to some fixed s ∈]1, 2] is c-lineable and latticeable in C([0, 1]) (see [6]).
Using the same notation as for f define the operator f : l ∞ → Bd([0, 1]) by Then f is well-defined, 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.
It therefore suffices to show that the upper Box-Counting dimension 1 (see [16]) fulfills dim B ( ( f (s))) ≤ α which can be done as follows: According to [16] in the calculation of the box-counting dimension it suffices to work with δ k = M 2 k meshes and k ∈ N. Since the set can be covered by one square of side length δ k , the minimum number N δ k ( f (s)) of squares of side length δ k needed to cover ( f (s)) fulfills 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 f is not an isometry and we can not simply conclude that the subspace f (l ∞ ) is closed. 1 Given any non-empty subset M of R n , we let N δ (M) denote the smallest number of sets, of diameter at most δ, needed to cover M. Then

Theorem 2.7 The family of all functions f ∈ C([0, 1]) whose graph has Hausdorff-and Box-Counting dimension equal to some fixed s ∈]1, 2] is c lineable and latticeable in
Focusing exclusively on the Hausdorff dimension, working with f : [1,2], and again using some bi-Lipschitz argument together with countable stability of the Hausdorff dimension even yields spaceability: The family of all functions f ∈ C([0, 1]) whose graph has Hausdorff dimension equal to some fixed s ∈]1, 2] is spaceable and latticeable in ( . Considering yet another small modification of f allows for quick alternative proofs for some of the results going back to [18,19]. In fact, settinĝ Proof (i) The assertion concerning 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 : 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 functionˆ f (s) is obviously differentiable on (0, 1]. Considering that the very (ii) The assertion concerning D ¬R follows in the same fashion.
We now turn to the family M s ⊆ C([0, 1]) of all functions f fulfilling that the sets U h , U h , defined by both have Hausdorff dimension s ∈]0, 1[ and, again using f for some appropriately chosen f , show that 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].
Proof Since the assertion is trivial for s ∈ {0, 1} it suffices to consider s ∈ (0, 1). Fix β ∈]0, 1 2 [ and define the contractions 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 non-empty compact subset C * β of [0, 1] fulfilling To simplify notation we will write W(K ) = w 1 (K ) ∪ w 2 (K ) for every non-empty compact subset K of [0, 1] and refer to 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 * β is self-similar and its it is straightforward to verify that T β is well-defined and a contraction on (F, d ∞ ), so Banach's Fixed Point Theorem implies the existence of a unique, globally attractive fixed point, which is easily verified to coincide with g * .

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 one-dimensional bounded intervals. Thus, fixed an integer m ≥ 0 and a real number 1 ≤ p < ∞ we consider, as vector space X , the Sobolev Space W m, p (]0, 1[). Before recalling basic facts about Sobolev spaces over intervals we start with some preparations which will be used subsequently.
Let (t n ) n∈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 Since binary sequences are identified with Z 2 numbers, the number t = ∞ n=0 t n 2 n associated to the sequence t n is the unique fixed point of the contraction For each fixed integer m ≥ 1, we will use the following functions in R [0,1] : In addition, for each j ∈ {1, . . . , m}, we recursively define the function s m, j ∈ R [0,1] by In the following lemma we collect some elementary properties of the functions s m, j . 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 ∈ N and 1 ≤ p ≤ ∞ the Sobolev space W m, p (]a, b[) can be defined as follows

Lemma 3.1 For every integer m ≥ 1 the function s m,m is non-negative and
where C ∞ c (]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 j-order of the function u and is denoted as usual by g j ≡ u j) .
The standard norm in the Sobolev space W m, p (]a, b[) is given by where · p denotes the usual L p -norm.  ([a, b]). Moreover the embedding is compact for 1 < p ≤ ∞. , b[) and satisfies the following properties: The main result in this section is the following.

Theorem 3.4 For every
Proof Fixed m and p, we consider the function f n ∈ R [0,1] defined by We observe that f n has derivatives up to order m and f Now we consider, for every d ∈ N, an infinite subset A d ⊂ N such that N = d∈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 γ d ∈ R [0,1] by where a n := n ln 2 (n + 1) −1/ p .
We claim that γ d ∈ W m, p (]0, 1[) \ q> p W m,q (]0, 1[). In order to prove the claim we first show that γ d admits weak derivatives up to order m. Indeed, given a function ϕ regular enough with supp ϕ∩] 1 2 d(n) , 1 2 d(n)−1 [ = ∅, we have, using the properties of f d(n) , that In particular, given ϕ ∈ C ∞ c (]0, 1[) we deduce that Since γ d and its derivatives up to order m − 1 are bounded functions we have that γ (2 d(n) x − 1)2 d(n)/ p a n , for every x ∈] In order to prove that H is closed we assume that g r ∈ H and that lim r →∞ g r −g m, p = 0 for some g ∈ W m, p (]0, 1[). 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 g r − g p ≤ g r − g m, p , we have that For each d ∈ N there exists a unique coefficient c d r ∈ R such that, given n ∈ N we have that Therefore it follows that Now we claim that, fixed d, the sequence c d r is bounded. Otherwise we may assumeup 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 ω > 0 such that This implies, using that the measure of S ω is positive, that This is a contradiction and this completes the proof that c d r is bounded. In addition, we may assume that -up to a subsequencec d r converges to some c d ∈ R. In particular which implies that T is well defined and even more, that l ∞ is continuously embedded in H .
In particular, H contain c independent vectors. We finally show that H is a lattice. Indeed, given z, . Observe that, for ϕ regular enough we have In particular, and, for j ∈ {1, . . . , m}, The properties of functions k m, j are analogous to those of s m, j and are collected in the following lemma. The main result about latticeability on Sobolev Spaces over unbounded intervals is the following.
where, as before, a n := n ln 2 (n + 1) Arguing as in the previous section the coefficients c e associated to any function ρ ∈ B are uniquely determined and B is a vector space with B\{0} ⊂ W m,  For each e ∈ N there exists a unique coefficient c e r ∈ R such that, given n ∈ N we have that