Classical vs. non-Archimedean analysis: an approach via algebraic genericity

In this paper, we show new results and improvements of the non-Archimedean counterpart of classical analysis in the theory of lineability. Besides analyzing the algebraic genericity of sets of functions having properties regarding continuity, discontinuity, Lipschitzianity, differentiability and analyticity, we also study the lineability of sets of sequences having properties concerning boundedness and convergence. In particular we show (among several other results) the algebraic genericity of: (i) functions that do not satisfy Liouville’s theorem, (ii) sequences that do not satisfy the classical theorem of Cèsaro, or (iii) functionals that do not satisfy the classical Hahn–Banach theorem.


Introduction and preliminaries
Throughout this paper, we are concerned with the study of rich algebraic structures within families of functions and sequences that are non-linear. This kind of study belongs to the area of lineability theory (intruduced by V. I. Gurariy in the early 2000's [6,31,45], and recently introduced by the AMS under classifications 15A03 and 46B87).
The one result that, likely, inspired the introduction of this theory was perhaps that of Levine and Milman [39] in 1940, which states that the set of all functions of bounded variation on [0, 1] does not contain a closed infinite dimensional subspace in C([0, 1]) endowed with the supremum norm.
Before carrying on with the main results of this manuscript, let us gather some necessary definitions and results from lineability theory. We shall use standard set-theoretical notation. As usual, P, N, N 0 , Z, Q, R and C denote the sets of all prime numbers, natural, natural numbers including zero, integer, rational, real, and complex numbers, respectively. We identify each cardinal number with the first ordinal of the same cardinality (thus, a cardinal κ is equal to the set of all ordinals less than κ, denoted κ = {α : α < κ}). Also, ℵ 0 and c denote the cardinalities of N and R, respectively. The restriction of a function f to a set A will be denoted by f A.
We shall frequently use the Fichtenholz-Kantorovich-Hausdorff theorem about families of independent sets in our proofs. This states that for any infinite set X there exists a family Y ⊆ P(X ) (called a family of independent sets) of cardinality 2 card(X ) such that for any finite sequences Y 1 , . . . , Y n ∈ Y and ε 1 , . . . , ε n ∈ {0, 1} we have Y ε 1 1 ∩ · · · ∩ Y ε n n = ∅, where Y 1 = Y and Y 0 = X \ Y , the complement of the set Y denoted Y c . Moreover, all the sets in Y can be chosen with cardinality card(X ); for more information see, for example, [26,34]. Here P(X ) denotes the power set of X . In what follows we fix P, N and B for a family of independent subsets of P, N and [0, 1], respectively. Now, we recall some usual terminology from the lineability theory. We say that a subset A of a vector space V over a field K is: If, in addition, V is a topological vector space over K, then A is said to be: Finally, and following [5,8], if V is a topological vector space contained in a (not necessarily unital) algebra and if κ and β are any (finite or infinite) cardinal numbers, then A is called: • κ-algebrable if there exists an algebra M such that M \{0} ⊆ A and M is a κ-dimensional vector space. Here, by S = {s α : α ∈ I } is a minimal system of generators of M, we mean that M is the algebra generated by S and for every α 0 ∈ I , s α 0 does not belong to the algebra generated by S \ {s α 0 }.
• strongly κ-algebrable if there exists a κ -generated free algebra M such that M \{0} ⊆ A.
Recall that an algebra M is called a κ-generated free algebra if there exists a subset X = {x α : α < κ} of M such that any function f from X to some algebra A can be uniquely extended to a homomorphism from M into A. Then X is called a set of free generators of the algebra M. In a commutative algebra we have a simple criterion; namely, a subset X = {x α : α < κ} in a commutative algebra B generates a free subalgebra M if and only if for any polynomial P without free term and any x α i ∈ X , 1 ≤ i ≤ n, we have P(x α 1 , . . . , x α n ) = 0 if and only if P = 0 . It should be noted that X = {x α : α < κ} ⊂ B is a set of free generators of a free algebra M ⊂ B if and only if the set of all elements of the form x k 1 α 1 x k 2 α 2 · · · x k n α n is linearly independent and all linear combinations of these elements (called algebraic combinations) are in B ∪ {0}. The notion of strong algebrability is essentially stronger than the notion of algebrability; see [8].
Recently, we began to study the lineability of certain subsets of functions and sequences over valued fields different from the classical fields of real and complex numbers; see [23,[36][37][38] . In this work we continue this study, investigating lineability and other related notions for the spaces of functions and sequences over the field of p-adic numbers in order to establish and compare the analogous of recent results within the context of non-Archimedean analysis. This ought to give a new insight in the theory of lineability in particular and in analysis in general by showing what properties of the scalar field are crucial for classical results.
This paper is arranged as follows. In Sect. 2, we recall some standard concepts and notations concerning non-Archimedean analysis. In Sect. 3, we first show that, if K ∈ {Q p , Q p , C p , p }, the set of everywhere surjective functions from K to K is 2 c -lineable, and (2 c , 2 c ) -algebrable when K = Q p ; generalizing [7] to the most important non-Archimedean fields. Some results about Darboux continuity and functions having continuity only at a fixed closed proper set are also given. Then we prove that the set of Lipschitz functions of order 1/m which are not of order 1/(m − 1) for any integer m ≥ 2 is c-lineable (obtaining as a consequence an improvement of [37] about the set of continuous nowhere differentiable functions). We also show that the set of discontinuous functions with finite range that have antiderivative and the set of separately continuous functions from Q n p to Q p with p > 2 and n ≥ 2 that are everywhere continuous except at a point are c-lineable. In Sect. 4, we begin by showing that the set of functions on Q p that do not satisfy Liouville's theorem is c-lineable. Then we prove a similar result for the set of sequences of functions that do not satisfy a well known classical theorem on the interchange of limit and derivative, and another result involving continuity and differentiablity. We also study the lineability of the set of non-locally constant functions that have derivative 0 with additional Lipschitz conditions. Finally, in Sect. 5, we provide some results on the spaces of p-adic sequences. More specifically, we show that the set of bounded sequences not converging to zero is c-spaceable and, within the same set, the family of sequences that also have only finitely many zero coordinates is strongly c-algebrable. We also prove the strong c-algebrability of the set of non-absolutely convergent series that are convergent; a result without any counterpart in the real case. In the same line of study, we also prove that the set of convergent sequences that are not Cesàro summable is strongly c-algebrable. Finally, we establish that the family of functionals on c 0 , considered over any non-spherically complete non-Archimedean field with non-trivial valuation, that do not have any continuous extension on ∞ is c -lineable. That means that the set of functionals that do not satisfy the classical Hahn-Banach theorem in the non-Archimedean setting is algebraically generic. Comparing with the classical cases, these results require an entirely new approach for their proofs.

A brief background p-adic analysis
We refer the interested reader for a more profound treatment of these topics to [4,30,35,43,44,46]. Now we turn to the main object of the paper; p-adic analysis. A non-Archimedean field is a field K equipped with a function (valuation) such that: • |x| = 0 if and only if x = 0, • |x y| = |x||y|, and • |x + y| ≤ max{|x|, |y|} (the strong triangle inequality), for all x, y ∈ K. Clearly, |1| = | − 1| = 1 and the valuation of summing n-times 1 is less or equal than 1 for all n ∈ N. An immediate consequence of the so-called strong triangle inequality is that |x| = |y| implies |x + y| = max{|x|, |y|}. A trivial example of a non-Archimedean valuation is the function | · | taking everything except 0 into 1 and |0| = 0. This valuation is referred to as the "trivial" one. By Big Ostrowski's theorem (see [46, theorem 1.2]) any complete valued field K that is not topologically isomorphic to R or C is non-Archimedean.
Let us denote by p an arbitrary prime number throughout this work. For any non-zero integer n = 0, let ord p (n) be the highest power of p which divides n. Then we define |n| p = p −ord p (n) , |0| p = 0 and | n m | p = p −ord p (n)+ord p (m) . The completion of the field of rationals, Q, with respect to the p-adic metric d(x, y) = |x − y| p is called the field of p-adic numbers Q p . The metric d satisfies the strong triangle inequality |x ± y| p ≤ max{|x| p , |y| p }. Ostrowski's theorem states that every non-trivial absolute value on Q is equivalent (i.e., defines the same topology) to an absolute value | · | p , for some prime number p, or the usual absolute value (see [30]).
Let a ∈ Q p and r be a positive number. The set B(a, r ) = {x ∈ Q p : |x − a| p < r } is called the open ball of radius r with center a, B(a, r ) = {x ∈ Q p : |x − a| p ≤ r } the closed ball of radius r with center a, and S(a, r ) = {x ∈ Q p : |x −a| p = r } the sphere of radius r and center a. The ring of integers in Q p is denoted by Z p , i.e., Z p = {x ∈ Q p : |x| p ≤ 1}. Note that every x ∈ Z p can be expanded in canonical form as x = a 0 + a 1 p + · · · + a k p k + · · · , a k ∈ {0, 1, . . . , p − 1}, k ≥ 0. We know that Z p is a compact set and N is dense in Z p [30]. Note that Q p is an infinite dimensional vector space over Q. In view of the fact that card(Z p \ {0}) = c and by applying the canonical representation of the p-adic rationals, we can take a Hamel basis of Q p over Q contained in Z p \ {0}.
As usual, we also denote by Q p the algebraic closure of Q p and by C p the completion of Q p with respect to the extended p-adic valuation. Finally, by p we denote the spherically complete extension of C p ; see [43,III.2]. A metric space is called spherically complete if each nested sequence of balls has a non-empty intersection.
Let us remark that the derivative of p-adic functions and analyticity of functions is defined as in the case of classical real functions, for more details, see [46]. Other functions that are relevant for our purposes are the Lipschitz functions. For any α > 0, the space of Lipschitz functions from K 1 to K 2 of order α is defined as The binomial coefficient functions x n are defined for x ∈ Z p and n ∈ N 0 by Finally, we define K-normed spaces. Let V be a linear space over a field K with a non-Archimedean non-trivial valuation | · |. A function · : V → [0, ∞) is said to be a non-Archimedean norm on V if the following conditions hold: Then (V , · ) is called a non-Archimedean normed space or a normed space over K. When V is complete with respect to the norm · , it is called Banach space over K.
The Banach space of all sequences (x n ) n≥0 with coordinates in K such that (x n ) n≥0 ∞ := sup n∈N 0 |x n | ≤ M, for some M ≥ 0, is denoted by ∞ . The subspace of ∞ consisting of all sequences converging to zero is denoted by c 0 . Unlike the Archimedean world, the dual space of c 0 is isometrically isomorphic to ∞ (see [42, theorem 2.5.11]).
Throughout this article we shall consider all vector spaces and algebras taken over the field K = Q p (unless stated otherwise).

Algebraic genericity of sets of p-adic discontinuous, continuous, and Lipschitz functions
We begin this section with a result about everywhere surjective functions. For a topological space X and a non-empty set Y , we say that f : X → Y is everywhere surjective (ES) provided f [U ] = Y for every non-empty open subset U of X . Apparently, the first example of these surprising functions on the real line is due to H. Lebesgue. Algebrability of these functions and other variants have been studied; see for example [7,9]. We give an optimal result in the sense of cardinality for the considered valued fields. To do this let us consider the following partition of Q p into c-many sets of cardinality c. For every α ∈ [0, 1], define Now, partition A α into c-many sets of cardinality c as follows: for every β ∈ [0, 1], define Also let us define for every α ∈ [0, 1] the function r α : Q p → K as follows: for every x ∈ Q p , The functions r α are ES. Indeed, fix a ∈ Q p and ε ∈ {p n : n ∈ Z}. It is enough to prove that r α [B(a, ε)] = K. Take y ∈ K arbitrary, then there exists a unique β ∈ [0, 1] such that γ (β) = y. Now choose x ∈ B(a, ε)\{a}, then 0 < |x −a| p = p t < p n for some t ∈ Z\{0}.
Case (1): If t > 0, then change the coefficients of p k where k ≥ 0 in the canonical representation of x so that x ∈ A α,β . Case (2): If t ≤ 0, then change the coefficients of p k where k > t in the canonical representation of x so that x ∈ A α,β .
Notice that in both cases we have x ∈ B(a, ε) ∩ A α,β . Hence, r α (x) = y. For every B ∈ B, we define r B = α∈B r α . We will prove first that the functions r B are well defined for any B ∈ B. For every x ∈ Q p , there exist unique α, β ∈ [0, 1] such that x ∈ A α,β . If α ∈ B, then r B (x) = γ (β) since {A α : α ∈ Q p } forms a partition of Q p and {A α,β : β ∈ Q p } forms a partition of A α . If not, then r B (x) = 0. Clearly, the function r B is ES for every B ∈ B since for any α ∈ B and any non-empty open subset U of Q p we have Assume now that K = Q p . Let us prove first that r B 1 does not belong to the algebra generated by {r B : B ∈ B}\{r B 1 }. Assume otherwise, that is, there exist B 2 , . . . , B n ∈ B\{B 1 } distinct and a polynomial P in n − 1 variables with coefficients in K \ {0} and without free term such that r B 1 = P(r B 2 , . . . , r B n ). Take α ∈ B 1 ∩ B c 2 ∩ · · · ∩ B c n , then for every x ∈ A α we have where P 1 is a polynomial in 1 variable with coefficients in K \ {0} and without free term. Therefore, as K \ {0} is algebraically closed, for every non-empty open set U of Q p we have

This proves (ii).
A consequence of Theorem 3.1 is the following.
Proof Fix K ∈ {Q p , C p , p } and let r B : Q p → K be the functions defined in the proof of Theorem 3.1. Take H be a Hamel basis of K over Q p containing 1. Then, for every x ∈ K, we have that x can be decomposed as Clearly the family of functions {ρ B : B ∈ B} is linearly independent. Now take a non-empty open subset U of K and fix x = β + y ∈ U , then there exists ε > 0 such that Assume now that ρ B 1 belongs to the algebra generated by {ρ B : B ∈ B}\{ρ B 1 }, then there exist B 2 , . . . , B n ∈ B \{B 1 } distinct and a polynomial P in n −1 variables with coefficients in K\{0} and without free term such that Finally, take B 1 , . . . , B n ∈ B distinct and P a polynomial in n variables with coefficients in K \ {0} and without free term such that where P 1 is a polynomial in 1 variable with coefficients in K \ {0} and without free term. Let U be a non-empty subset of K and take an arbitrary open ball B(x, ε) ⊂ U , where x = β + y. Therefore, as K \ {0} is algebraically closed, we have Let us recall that a subset C of a non-Archimedean field K is called convex if λx+μy+νz ∈ C for every x, y, z ∈ C, and λ, μ, ν ∈ K with |λ| , |μ|, |ν| ≤ 1 and λ + μ + ν = 1 (see [42, theorem 3.1.15]). It can be seen that the only convex subsets of K are ∅, K, the singleton sets and balls (see [42, p. 89

Remark 3.3
Notice that the definition of convex set over a non-Archimedean field K is not the classical definition of a convex set in the Archimedean fields R or C. Recall that a subset C of R (resp. C) is convex provided that λx + μy ∈ C for every x, y ∈ C, and λ, μ ∈ R (resp. C) with |λ| , |μ| ≤ 1 and λ + μ = 1. The reader may think that such definition can be adapted to any non-Archimedean field but this is not the case, for instance, on Q 2 , since the residue class field of Q 2 is the finite field of 2 elements F 2 (see [42, is an ES function and C is a convex subset of Q p (resp. K), then f [C] is the empty set, a singleton set, or K, i.e., a convex subset of K. Hence, as an immediate consequence of Theorem 3.1 and Corollary 3.2, we have the following corollary, which generalizes [38, theorem 2.3].
Darboux continuous but not continuous is Proof For every n ∈ N 0 , we denote by S n the following set: are open and, therefore, h α is locally constant for every α ∈ (0, 1). Hence, h α is continuous for every α ∈ (0, 1) and, thus, every algebraic combination of the functions {h α : α ∈ (0, 1)} is also continuous. Take, without loss of generality, 0 < α 1 < · · · < α k < 1. Let P be a polynomial in k variables with coefficients in Q p \ {0} and without free term. If P i are the monomials that form P and x ∈ S n , then In particular, where γ i ∈ Q p \ {0} are the coefficients of P and i k , . . . , i 1 are non-negative integers. Assume, without loss of generality, that the k-tuples (i k , . . . , i 1 ) from (3.1) appear ordered lexicographically. Notice that all the k-tuples are distinct since otherwise we could add the monomials in the polynomial P that have these k-tuples as exponents. Clearly, we have that is the only k-tuple). Hence, there exists n 0 ∈ N such that for any n ≥ n 0 we have In [38, proposition 2.1], Khodabendehlou and the second and fourth authors prove that given a closed proper subset F of Q p , the family of functions Q p → Q p that are continuous only at the points belonging to F is 2 c -lineable. Here we give a strong version of the result. Theorem 3.6 If K ∈ {Q p , C p , p } and F is a closed proper subset of K, then the family of functions K → K that are continuous only at the points belonging to F is (2 c , 2 c )-algebrable.
Following the proof of Theorem 3.1 define the functions ϕ B = α∈B t α for every B ∈ B.
As in the proof of Theorem 3.1, notice that the functions ϕ B are well defined functions from Take H a Hamel basis of K over Q p containing 1. Then, for every x ∈ K, we have that x can be decomposed as Consider now the distance function d (x, F) = min |x − c| p : c ∈ F , where x ∈ K, and take the sets Now define the function d(·, F) : K → Q p as follows: We will prove that the family of functions {d B : B ∈ B} are continuous only at the points belonging to F. Take x ∈ F, i.e., d B (x) = 0, and let (x n ) n≥1 ⊂ K be a sequence converging to x, then (d B (x n )) n≥1 converges to 0. Indeed, notice that for every n ∈ N we have F is continuous. Now take x / ∈ F. By means of contradiction, assume that d B is continuous at x. We have two cases: converges to x ∈ F m for some m ∈ N 0 , we have that there exists n 0 ∈ N such that |d B (x n )| p ≥ p −m−1 for every n ≥ n 0 . Hence, (d B (x n )) n≥1 cannot converge to 0, a contradiction. Case (2): Assume that φ B (x) = 0. Once again, as φ B is ES from K to Z p , we can choose a sequence (x n ) n≥1 ⊂ F c converging to x such that φ B (x n ) = 0 for every n ∈ N. Since x ∈ F m for some m ∈ N 0 , we have that d B (x) = 0. But d B (x n ) = 0 for every n ∈ N, a contradiction.
Thus d B is not continuous at the points outside F. Let B 1 ∈ B, we will prove that d B 1 does not belong to the algebra generated by {d B : B ∈ B} \ {d B 1 }. Assume that d B 1 can be written as P(d B 2 , . . . , d B n ), where B 2 , . . . , B n ∈ B \ B 1 are distinct and P is a polynomial in n − 1 variables with coefficients in K \ {0} and without free term.
Analogously to the proof of Corollary 3.2 we have that given It is clear that any algebraic combination of the functions {d B : B ∈ B} over K is continuous at the points belonging to F . It remains to show the following: Given B 1 , . . . , B n ∈ B distinct and assuming that P is a polynomial in n variables with coefficients in K \ {0} and without free term such that P(d B 1 (x 0 ), . . . , d B n (x 0 )) = 0 for some x 0 ∈ K \ {0}, then P(d B 1 , . . . , d B n ) is discontinuous outside F. Now, there exists α ∈ B 1 ∪ · · · ∪ B n such that x 0 = β 0 + y 0 with β 0 ∈ A α and y 0 ∈ span H \ {1} , since otherwise P(d B 1 (x 0 ), . . . , d B n (x 0 )) = P(0, . . . , 0) = 0. Moreover, x 0 ∈ F c , if not, then P(d B 1 (x 0 ), . . . , d B n (x 0 )) = P(0, . . . , 0) = 0. Hence, for any x = β + y ∈ F c with β ∈ A α , we have that P(d B 1 (x), . . . , d B n (x)) is of the form P 1 (t α (β)d(x, F)), where P 1 is a polynomial in 1 variable with coefficients in K \ {0} and without free term. Fix x ∈ F m for some m ∈ N 0 . If m = 0, take V x a neighborhood of x sufficiently small contained in F m ∪ F m−1 . If m = 0, take V x contained in F 0 . Notice that there exists a neighborhood U β of β such that U β + y ⊂ V x . We have two cases: Case (1): Assume that P(d B 1 (x), . . . , d B n (x)) = 0. Since P 1 takes the value 0 at most on a finite set C ⊂ K, take a sequence (x r ) r ≥1 that converges to x satisfying: x r = β r + y r ∈ V x with β r ∈ A α ∩U β and y r ∈ span H \{1} , (t α (β r )) r ≥1 is a constant sequence, and (t α (β r )d(x r , F)) n≥1 ⊂ ∪ z∈C B(z, ε) with ε > 0 sufficiently small. This can be done as t α is ES from Q p to Z p and d(x r , F) ∈ p m , p m+1 for every r ∈ N. Hence, (P 1 (t α (β r )d(x r , F))) r ≥1 does not converge to 0. Case (2): Assume that P(d B 1 (x), . . . , d B n (x)) = 0. Take α ∈ B c 1 ∩· · ·∩ B c n and a sequence (x r ) r ≥1 converging to x such that x r = β r + y r ∈ V x with β r ∈ A α and y r ∈ span H \ {1} . Now we have that (P(d B 1 (x r ), . . . , d B n (x r ))) r ≥1 is the zero sequence which cannot converge to P(d B 1 (x), . . . , d B n (x)).
Thus, P(d B 1 , . . . , d B n ) is not continuous at any point of F c .
In the next result, we give a p-adic analogous of [3, theorem 5.1]. Here we need a more delicate argument in comparison to the real case. We equip the space Q n p with the following norm: Theorem 3.7 Let n ≥ 2 and x 0 ∈ Q n p with p > 2. The family of separately continuous functions Q n p → Q p that are everywhere continuous except at x 0 is c-lineable.

Proof
We will prove the result in the case when n = 2. Fix x 0 = 0, where 0 := (0, 0). For any x, y ∈ Z p written in the canonical form x = ∞ n=0 a n p n and y = ∞ n=0 b n p n , let us denote c(x, y) := ∞ n=0 c n p n , where c n = a n and c 2n+1 = b n . Let us also denote x = ∞ n=1 a 2n+1 p 2n + a 2n p 2n+1 for any x = ∞ n=0 a n p n ∈ Z p . Let us define f : which is an open set and, therefore, for every 0 < ε < p r , we have Notice that we have the following On the other hand, for any k ∈ N, consider the sequence (x k n , y k n ) n≥0 ⊂ Z 2 p , where x k n = 2 p 2n + 2 p 2n+1 + 2 p 2n+2k+1 and y k n = p 2n + p 2n+1 + p 2n+2k+1 for every n ≥ 0. Then, for any n ≥ 0, we have that Thus, for any k ∈ N, when n → ∞ we have that (x k n , y k n ) n≥0 tends to zero in Q 2 p and f (x k n , y k n ) tends in Q p to the constant Notice that for distinct k, k ∈ N, the p-adic numbers β k , β k are p-integers and also distinct. Hence f is not continuous at 0. It is easy to see that f is separately continuous since f (0, x) = f (y, 0) = 1 for every x, y ∈ Q p . Let V be a family of c-many analytic linearly independent functions from Q p to Q p . For every v ∈ V, let us define the function where f v j are distinct and α j ∈ Q p for any 1 ≤ j ≤ n. If F were identically zero, then n j=1 α j v j (β k ) = 0 for every k ∈ N. Therefore, the function n j=1 α j v j (which is analytic) would be equal to zero on an infinite set which contradicts Strassman's theorem [30, theorem 4.4.6]. Notice that by construction the function F is continuous except at 0 and also separately continuous.
The next result shows that the inclusion is proper, with K 1 ∈ {Z p , Q p } and K 2 ∈ {Q p , Q p , C p , p }; and large enough to contain a c-dimensional linear space over K 2 .

Theorem 3.8 For every
and an integer m ≥ 2. Given q ∈ P we will define a function f q : K 1 → Z p in the following way: for every x = ∞ n=r a n p n , with r ∈ Z, take f q (x) = ∞ n=0 a m(m−1)q n+1 p (m−1)q n+1 . Let us prove that for any q ∈ P, we have f q ∈ Lip 1/m (K 1 → K 2 ). Take x, y ∈ K 1 .
Clearly | f q (x) − f q (y)| p ≤ |x − y| 1/m p when x = y, so assume that x = y. Hence there exists t ∈ Z such that |x − y| p = p −t . We will divide the proof into two cases: (2): If t > m(m − 1)q, then there exists n t ∈ N 0 such that m(m − 1)q n t +1 < t ≤ m(m − 1)q n t +2 . Therefore, and, hence, we have For every P ∈ P, let us define F P = q∈P f q . Fix P ∈ P. The function F P is well defined since, for every x ∈ K 1 , F P (x) exists. Indeed, it is enough to prove that ( f q (x)) q∈P converges to 0 for every x ∈ K 1 . Take x ∈ K 1 , then | f q (x)| p ≤ p −(m−1)q → 0 as q → ∞.
Let us prove now that F P / ∈ Lip 1/(m−1) (K 1 → K 2 ) for every P ∈ P. Fix P ∈ P. For every q ∈ P, take x = 0 and y i = p m(m−1)q i+1 for any i ∈ N 0 . Notice that |y i | p = p −m(m−1)q i+1 and |F P (y i )| = p −(m−1)q i+1 for any i ∈ N 0 . Assume that F P ∈ Lip 1/(m−1) (K 1 → K 2 ), for any i ∈ N 0 . Thus, we have p −(m−1)q i+1 ≤ Mp −mq i+1 for any i ∈ N 0 , that is, p q i+1 ≤ M for any i ∈ N 0 , which is absurd. Now we will show that F P ∈ Lip 1/m (K 1 → K 2 ) for every P ∈ P. For every x, y ∈ K 1 we have that Let us show that the functions in the set {F P : P ∈ P} are linearly independent over K 2 . Take n distinct elements P 1 , . . . , P n of P and consider F = n j=1 b j F P j with b j ∈ K 2 \ {0} for every 1 ≤ j ≤ n. Notice that by picking q ∈ P 1 ∩ P c 2 ∩ · · · ∩ P c n and x = p m(m−1)q , we have F(x) = b 1 p (m−1)q . If F were the identically zero function, then b 1 = 0 and we have a contradiction.
Since Lip 1/m (K 1 → K 2 ) forms a vector space over K 2 , we clearly have that any linear combination over K 2 of the functions in {F P : P ∈ P} belongs to Lip 1/m (K 1 → K 2 ). It remains to prove that given F = n j=1 b j F P j with b j ∈ K 2 \ {0} and P 1 , . . . , P n distinct elements of P, we have that F / ∈ Lip 1/(m−1) (K 1 → K 2 ). Take q ∈ P 1 ∩ P c 2 ∩· · ·∩ P c n , x = 0 and y i = p m(m−1)q i+1 for any i ∈ N 0 . Notice that |F(x) − F(y i )| p = |b 1 | p −(m−1)q i+1 for any i ∈ N 0 . By using the same arguments as above we have that In [37, theorem 4.5], the authors prove that the family of uniformly continuous functions Z p → Q p that are nowhere differentiable functions is c-lineable provided that p > 2. These functions can be named Dieudonné's monster in parallel to its counterpart in the real case named Weierstrass' monster.
By taking m = 2 in Theorem 3.8, note that the functions in the set Lip 1/2 (K 1 → K 2 ) \ Lip 1 (K 1 → K 2 ), with K 1 ∈ {Q p , Q p } and K 2 ∈ {Q p , Q p , C p , p }, are uniformly continuous and nowhere differentiable. Hence, we have the following result.

Corollary 3.9
For every K 1 ∈ {Z p , Q p } and K 2 ∈ {Q p , Q p , C p , p }, the family of uniformly continuous nowhere differentiable functions K 1 → K 2 is c-lineable (as a K 2 -vector space).
By an antiderivative of a function f we mean any function F such that F = f . We give a weaker corrected version of [38, theorem 2.7] . Let us remark that the functions constructed in its proof are not everywhere discontinuous and do not generate an algebra.

Proposition 3.10
The family of discontinuous functions Q p → Q p with finite range that have antiderivative is c-lineable.
Proof For every N ∈ N , let us define g N : Q p → Q p as For any N ∈ N , the function g N clearly has finite range and has an antiderivative G N given by Firstly, if x = 0 and |x| p / ∈ p −n : n ∈ N , there is a neighborhood U x of x such that G N U x is the identity function. Thus, G N is differentiable at x and the derivative is 1 = g N (x). Secondly, if |x| p ∈ p −n : n ∈ N , there is a neighborhood U x of x such that G N U x is constant. Hence, G N is differentiable at x and the derivative is 0 = g N (x). Lastly, we will analyze the case when x = 0. Notice that otherwise.

Lineability of sets of p-adic differentiable and analytic functions
We commence by showing the failure of celebrated Liouville's theorem on p -adic numbers field. Let us remark that Liouville's theorem, which states that a bounded analytical function of a field K is constant, holds true in any not locally compact complete non-trivially valued field [44, theorem 42.6]. But this is not true for the locally compact case as illustrated in [44, example 43.1]. We show that this is a generic algebraic behavior of bounded analytic functions on Q p .

Theorem 4.1 The family of non-constant bounded analytic functions
Proof First, it is important to mention that the function f given in the proof of [44, example 43.1] satisfies f (0) = 1. Also, by the proof of [44, example 43.1], we can assume that | f (x)| p ≤ p −1 . By considering nowf = f − 1, we have a non-constant bounded analytic function from Q p to Q p , which will be called again f for simplicity, that satisfies the following properties: f (0) = 0 and | f (x)| p ≤ 1 for any x ∈ Q p . Hence, since f is analytic, there exists (a n ) n≥0 ⊂ Q p such that f (x) = ∞ i=0 a i x i for any x ∈ Q p and, as f (0) = 0, notice that a 0 = 0.
Given q ∈ P, let us define the function g q (x) = p q f (x) for every x ∈ Q p . For every P ∈ P, take G P = q∈P g q . Fix P ∈ P. The function G P is well defined since G P (x) exists for every x ∈ Q p . Indeed, for every x ∈ Q p , we have that (g q (x)) q∈P converges to 0 since |g q (x)| p ≤ p −q → 0 as q → ∞. Moreover, the functions G P are bounded since for every x ∈ Q p we have Now we will prove that the functions G P are analytic in Q p . For every α ∈ Q \ Z, let us define a α = 0. Notice that the function i∈N q∈P p q a i q x i q is clearly analytic in Q p and for any integer n ≥ 2 and every x ∈ Q p we have On the other hand, as ∞ i=0 a i y i converges for any y ∈ Q p , the sequence (a i y i ) i≥0 converges to 0 for any y ∈ Q p (see, for instance, [35, proposition 3.3]). Hence, there exists A y > 0 such that |a i y i | p ≤ A y for any i ∈ N 0 . Thus It is clear that any non-zero linear combination on Q p of functions in the set {G P : P ∈ P} is a non-constant, bounded and analytic function from Q p to Q p . It remains to prove that they are linearly independent over Q p . Let i 0 ∈ N be the index such that a i 0 = 0 and a i = 0 for every 0 ≤ i ≤ i 0 −1. Take n distinct sets P 1 , . . . , P n in P and the function G = n j=1 b j G P j with b j ∈ Q p \ {0}. Notice that by taking q ∈ P 1 ∩ P c 2 ∩ · · · ∩ P c n , the coefficient of x i q with i = i 0 q in the power series expansion of G P is b 1 p q a i 0 . Hence, if G were identically zero, we would have b 1 = 0, which is absurd.
In the next proposition we study the failure of one of the standard results in real analysis about the interchange of limit and derivative. More specifically, as K. Mahler put it [40]: if a series f (x) = n f n (x) converges and the derived series g(x) = n f n (x) converges uniformly, g(x) still need not be the derivative of f (x). To do that, we need the van der Put expansion of a continuous function on Z p . For an integer m > 0 and x ∈ Z p define and ψ 0 (x) ≡ characteristic function of the ball B(0, 1/ p).
It is a well known result of van der Put that every continuous function f : where α m ∈ K; see [41,43,44]. It should be noted that the series converges uniformly, and the functions ψ m form a basis of the space of locally constant functions Z p → K (see [43, pp. 179-182]). Now we are ready to prove our next result.

Theorem 4.2
For every K ∈ {Q p , Q p , C p , p }, the set of functions f : Z p → K such that there exists a sequence of differentiable functions ( f n ) n≥1 with f n : Z p → K, f n → f uniformly and f n → g uniformly but f = g is c-lineable.
Proof Fix K ∈ {Q p , Q p , C p , p }. For every N ∈ N , let us define the power series f N (x) = n∈N p n 2 x n for any x ∈ Z p . Clearly, the radius of convergence of f N is infinite for any N ∈ N . Now, let f = a 1 f N 1 + · · · + a k f N k , where k ∈ N, N 1 , . . . , N k ∈ N are distinct and a 1 , . . . , a k ∈ K \ {0}. Notice that f is a power series with coefficients in K. If f were the zero power series, then the coefficients of f in the terms x n with n ∈ N 1 ∩ N c 2 ∩ · · · ∩ N c k , which are a 1 p n 2 , would be zero. Thus, we would have a 1 = 0 which is a contradiction. An analogous approach shows that the derivative of f given by f = a 1 f N 1 + · · · + a k f N k , where f N i (x) = n∈N i np n 2 x n−1 , is not the zero power series.
Let F = { f N : N ∈ N }. For any f N ∈ F and n ∈ N 0 , take f n,N to be the partial sum n i=0 α i ψ i of the van der Put expansion of the function f N . On the one hand, as noted above, the sequence ( f n,N ) n≥0 converges uniformly to f N for any N ∈ N . On the other hand, in view of the fact that (ψ i ) i≥0 is a basis of the space of locally constant functions Z p → K, we have that f n,N ≡ 0 for any n ∈ N 0 and N ∈ N . Hence, ( f n,N ) n≥0 converges uniformly to the zero function for any N ∈ N . But f N ≡ 0 for any N ∈ N . The same can be applied for any non-zero linear combination over K of the functions in F .
It is well known that if f : R → R is differentiable with f ≡ 0, then f is constant. This is not true in general in the p-adic setting (see [35, example 4.26]). In [37], it was shown that the set of functions Q p → Q p with f ≡ 0 that are not constant (or locally constant) on any ball is c-lineable. We show that this can be improved by restricting ourselfs to the Lipschitzian functions. It should be noted that in real analysis, the Lipschitz functions of order α > 1 are trivial.

Theorem 4.3
For every K ∈ {Z p , Q p }, the family of non-locally constant functions f : K → Q p whose derivative is the zero function and f belongs to Lip α (K → Q p ) for every α > 0 is c-lineable.
Proof Fix K ∈ {Z p , Q p }. Given q ∈ P, let us define the function f q : K → Z p as: for every x = ∞ n=r a n p n , with r ∈ Z, we have f q (x) = ∞ n=0 a n p q (n+1)! . For every P ∈ P, define F P : K → Z p by F P = q∈P f q . Once again we have that F P is well defined for every P ∈ P since, by fixing P ∈ P, the sequence ( f q (x)) q∈P converges for every x ∈ Z p . Indeed, we have f q (x) p ≤ p −q → 0 as q → ∞ for any x ∈ K.
We will prove that the set {F P : P ∈ P} is a family of linearly independent functions over Q p such that any non-zero linear combination of these functions over Q p is non-locally constant, belongs to Lip α (Z p → Q p ) for every α > 0 and its derivative is the zero function.
It is clear that the functions F P and any non-zero linear combination of these functions is non-locally constant. Now we will show that they belong to Lip α (K → Q p ) for any α > 0. Fix α > 0. For any distinct x, y ∈ K such that |x − y| p = p t for some t ∈ Z we have that |F P (x)−F P (y)| p |x−y| α p = p −q (t+1)! +αt → 0 as t → ∞, and whereq = min{q ∈ P}. Hence, notice that the latter proves that F P ∈ Lip α (K → Q p ), and in the case α = 1 we have that F P ≡ 0. Now let F = n j=1 b j F P j , where n ∈ N, b j ∈ Q p \ {0} for any 1 ≤ j ≤ n and P 1 , . . . , P n are n distinct elements of P. Assume that F is the zero function, then by fixing q ∈ P 1 ∩ P c 2 ∩ · · · ∩ P c n we have that F( p q ) = b 1 p q (q+1)! = 0 if and only if b 1 = 0, a contradiction. Now, it is easy to see that F ∈ Lip α (K → Q p ) for any α > 0 and F ≡ 0.
In [41, p. 200], by using Mahler series, Mahler constructs a continuous function that is not differentiable only at a point but has continuous derivative elsewhere. By a totally different example, we show that the set of such functions is c-lineable. Proof For any q ∈ P, let us define f q : Z p → Z p in the following way: The function f q is locally constant at every point except 0, i.e., for every x ∈ Z p , there exists a neighborhood V x of x such that f q V x is constant. Indeed, let x ∈ Z p , then we have two cases: Case (1): If x ∈ S 0, p −q n+1 for some n ∈ N, then f q S 0, p −q n+1 ≡ p q n .
Case (2): If x / ∈ S 0, p −q n+1 for every n ∈ N, then x ∈ S 0, p −k for some k ∈ N 0 \ p −q n+1 : n ∈ N . Applying the same arguments as in Case (1) we have that f q is identically zero on some neighborhood of x.
Moreover, f q is not locally constant at 0 since for every ε ∈ {p −k : k ∈ N 0 }, there exists distinct n, m ∈ N such that p −q n+1 < ε and p −q m+1 < ε. Hence, the spheres S 0, p −q n+1 and S 0, p −q m+1 are contained in B(0, ε), which shows that f q takes at least two distinct values in B(0, ε). (In fact, it takes infinitely many values.) Therefore, we have proven that f q is differentiable at every point except maybe 0 (we will see later that it is not differentiable at 0). However, f q is continuous at 0 (obviously it is continuous at every other point x = 0 since f q is differentiable at x = 0). Indeed, let ε ∈ {p −k : k ∈ N 0 } and take n ∈ N 0 such that p −q n < ε. For any as n → ∞.
For every P ∈ P, define the function F P = q∈P f q . Using similar arguments used before, we see that F P is well defined. Moreover, by applying similar arguments used to prove that f q is continuous everywhere and differentiable at every point x = 0, we have that F P is differentiable at every point x = 0 and hence continuous at every point x = 0. But also continuous at 0 since F P is the uniform limit of the sequence of continuous functions q∈P q≤k f q k≥2 . Furthermore, the functions F P are not differentiable at 0. To see this, fix q ∈ P. Then, Also the functions F P are linearly independent over Q p (apply similar arguments used in other proofs of this work). Finally, any non-zero linear combination over Q p of the functions F p satisfies the desires properties. Indeed, take r ∈ N distinct elements of P. Namely, P 1 , . . . , P r , and take F = n j=1 a j F P j , where a j ∈ Q p \ {0}. Clearly F is continuous everywhere and differentiable at every point x = 0. Now, fix q ∈ P 1 ∩ P c 2 ∩ · · · ∩ P c r . Notice that F( p q n+1 ) = a 1 p q n = 0, then as n → ∞, which shows that F is not differentiable at 0.

p-adic sequence spaces and failure of the Cesàro and Hahn-Banach theorems
In this section we present some results about lineability, algebrability and spaceability of some subsets of the space of p-adic sequences and conclude with a result concerning the failure of the Hahn-Banach theorem in the p-adic setting. To begin, we give an improvement of [36, proposition 2.1].
Theorem 5.1 Let K be a non-Archimedean field with non-trivial valuation. If ∞ and c 0 are defined over K, then the set ∞ \ c 0 is c-spaceable.
Proof For every N ∈ N , let us define the sequence x N as follows: for every n ∈ N, Notice that the sequences in the set {x N : N ∈ N } are linearly independent over K since N is a family of independent subsets of N. Take V = span {x N : N ∈ N } . Clearly, any x ∈ V is bounded. Also, if x is not the zero sequence, then x does not converge to 0. Indeed, assume that x = m i=1 a i x N i , where a i ∈ K \ {0} and N i ∈ N for every 1 ≤ i ≤ m. Then, for every n ∈ N 1 ∩ N c 2 ∩ · · · ∩ N c j we have x(n) = a 1 , i.e., x restricted to the infinite set N 1 ∩ N c 2 ∩ · · · ∩ N c m is a constant non-zero infinite sequence. Now, let x ∈ V \ {0}, then there exists (s k ) k≥1 ⊂ V \ {0} converging (uniformly) to x, i.e., s k − x ∞ → 0 as k → ∞. We will prove that x ∈ ∞ \ c 0 . Clearly, x is bounded since (s k ) k≥1 ⊂ ∞ . As x is not the zero sequence, there exists n 0 ∈ N such that x(n 0 ) = 0, that is, there exists r ∈ (0, ∞) such that |x(n 0 )| p = r . Thus, there exists k 0 ∈ N such that s k − x ∞ < r 2 for every k ≥ k 0 . For every k ≥ k 0 , let N 1,k , . . . , N m,k be the sets that form s k . Assume that n 0 ∈ N ε 1 1,k ∩ · · · ∩ N ε m m,k , where ε i ∈ {0, 1} for every 1 ≤ i ≤ m. Then, since N ε 1 1,k ∩ · · · ∩ N ε m m,k is infinite and the sequence s k restricted to N ε 1 1,k ∩ · · · ∩ N ε m m,k is a constant sequence, we have s k (n) = s k (n 0 ) for infinitely many n. Notice that Therefore, since |x(n 0 ) − s k (n 0 )| < r 2 , we have max{|x(n 0 ) − s k (n 0 )|, |s k (n 0 )|} = |s k (n 0 )|. Thus, |s k (n 0 )| ≥ r . Hence, |s k (n)| ≥ r for infinitely many n. The latter implies that for infinitely many n we have This proves that |x(n)| > r 2 > 0 for infinitely many n. Therefore, x does not converge to 0 in K and the proof is complete.
We pause to analyze the functions (1 + x) α where x ∈ pZ p and α ∈ Z p . By definition, the function α i x i . Notice that by construction (1 + x) α is a function from pZ p to Z p . The function (1 + x) α satisfies the following properties which can be found in [44, pp. 138-142]: (ii) For x fixed, the sequence n i=0 α i x i converges uniformly (see [43]). Hence, since pZ p is a compact metric space, the function F(x, α) is continuous in the second variable. (iii) As a consequence of (ii) and the fact that N is dense in Z p , we have Let us continue by proving the following lemma which will be very useful in the sequel.

Lemma 5.2
If α 1 , . . . , α n ∈ Z p \ {0} are distinct, with n ∈ N, then there is no linear Proof We will prove it by induction on n. For n = 1 we have that F(x, α) is a non-constant analytic function, and hence γ F(x, α) is not constant for every γ ∈ Q p \ {0}. We claim that the lemma is true up to n − 1. Now, let α 1 , . . . , α n ∈ Z p \ {0} and assume that for some γ i ∈ Q p \ {0} for every 1 ≤ i ≤ n, and γ ∈ Q p . We have two cases: Case (1): If γ = 0, then n i=2 γ i (1 + x) α i −α 1 = γ 1 for every x ∈ pZ p , which contradicts the inductive hypothesis (this can be done since −1 / ∈ pZ p ). Case (2): If γ = 0, then by differentiating (5.1) we have n i=1 γ i α i (1 + x) α i −1 = 0. On the one hand, if α i = 1 for every 1 ≤ i ≤ n, then we proceed as in Case (1). On the other hand, if α i = 1 for some 1 ≤ i ≤ n, then we would reach a contradiction with the inductive hypothesis.
Having these non-Archimedean tools, we are ready to show an improvement and generalization of [18, proposition 2.1] to the p-adic setting; see also [36, proposition 2.4].

Theorem 5.3
The subset of ∞ \ c 0 defined over Q p whose elements only have finitely many zero coordinates is strongly c-algebrable.
Proof Let {z n } n≥1 be an enumeration of Z and for any α ∈ Z p \ {0} consider the sequence ((1 + pz n ) α ) n≥1 . Take H a Hamel basis of Q p over Q contained in Z p \ {0} . Notice that the sequences {((1 + pz n ) h ) n≥1 : h ∈ H} are algebraically independent. Indeed, first of all, note that for every n ∈ N, the function P((1+ pz n ) h 1 , . . . , (1+ pz n ) h m ) (where P is a polynomial in m ∈ N variables with coefficients in Q p \ {0} and without free term, and h i ∈ H for every 1 ≤ i ≤ m) can be written as x being an arbitrary element of pZ p and, therefore, contradicting Lemma 5.2.
It is easy to see that the set of all conditionally convergent series of real numbers is algebrable with respect to the pointwise multiplication while, in [36], it was shown that in the p-adic setting the family of all sequences whose series is convergent but not absolutely convergent is (ℵ 0 , 1)-algebrable in c 0 . Here we prove a stronger and more optimal version of this result.

Theorem 5.4
In the space c 0 over Q p , the family of all sequences whose series is convergent but not absolutely convergent is strongly c-algebrable with respect to the pointwise multiplication.
Proof Let (r n ) n≥1 be the sequence of exponents in the sequence (t n ) n≥1 given in the proof of [36, proposition 4.2] and {z n } n≥1 be an enumeration of Z. For every α ∈ Z p \ {0}, consider the sequence ( p r n (1 + pz r n ) α ) n≥1 . Now, take H a Hamel basis of Q p over Q contained in Z p \ {0}. We will prove that the sequences in {( p r n (1 + pz r n ) h ) n≥1 : h ∈ H} are algebraically independent by showing that any linear combination s of products of the sequences ( p r n (1 + pz r n ) α ) n≥1 is not absolutely convergent.
For any n ∈ N, notice that s(n) is of the form where a ∈ N, b i ∈ N and γ i ∈ N with γ i < γ i+1 , α i, j ∈ Z p \ {0} and β i, j ∈ Q p with β i, j = 0 for some pair (i, j). First of all, s converges in Q p since z N (m) = p ln β(N ) (ln(n)) if m = p n with n ∈ N , 0 otherwise.
In fact, it is clear that T N = 1 for any N ∈ N . Hence, T N ∈ c 0 for any N ∈ N . Furthermore, the functionals T N are linearly independent. To see this, take T = k i=1 a i T N i , where a i ∈ K \ {0} and with N i ∈ N distinct for every 1 ≤ i ≤ k. Assume that T is the zero functional and fix n 0 ∈ N 1 ∩ N c 2 ∩ · · · ∩ N c k . Now consider x = (x n ) n≥1 defined by x n = 1 if n = n 0 , 0 otherwise.
Clearly, x ∈ c 0 . Moreover, 0 = T (x) = a 1 , and we have a contradiction. Let T = span T N : N ∈ N . If T ∈ T , then T clearly belongs to c 0 .

Lemma 5.7
Let K be a non-Archimedean field with non-trivial valuation. For every T ∈ span T N : N ∈ N \ {0}, the following properties on K are equivalent: (i) K is spherically complete.
(ii) The functional T ∈ c 0 can be extended to a functional T ∈ ( ∞ ) . It remains to prove (ii) ⇒ (iii). By (ii), let T ∈ ( ∞ ) be an extension of T . Let us denote N = N \ k j=1 N 0 j . Take (k n ) n≥1 ⊂ N the strictly increasing sequence such that N = {k n : n ∈ N}. Consider φ : N → N the bijection defined as φ(n) = k n , for every n ∈ N, and define the auxiliary operator R : ∞ → ∞ as follows: for every x = (x n ) n≥1 ∈ ∞ , where, for every n ∈ N, the coordinate y n is defined in the following way.
If n = k m for some m ∈ N with k m ∈ k j=1 N ε j j \ min k j=1 N ε j j , where ε 1 , . . . , ε k ∈ {0, 1} with ε l = 0 for some 1 ≤ l ≤ k, then Otherwise, y n = 0. Notice that R is a continuous linear operator on ∞ such that T • R − T = 0 on c 0 . Assume that ( ∞ /c 0 ) = {0}, then T • R = T on ∞ . For every x = (x n ) n≥1 ∈ ∞ , let us define inductively the sequence z = (z n ) n≥1 in the following way.
Take z 1 = x 1 . For n ≥ 2, assume that we have already defined z r for every 1 ≤ r ≤ n − 1. If n = k m for every m ∈ N, or n = k m for some m ∈ N with k m = min k j=1 N ε j j and where ε 1 , . . . , ε k ∈ {0, 1} with ε l = 0 for some 1 ≤ l ≤ k, then z n = x n . If n = k m for some m ∈ N with k m ∈ k j=1 N ε j j \ min k j=1 N ε j j , where ε 1 , . . . , ε k ∈ {0, 1} with ε l = 0 for some 1 ≤ l ≤ k, then z n = x k m + z max k s ∈ k j=1 N ε j j : k s <k m .
Since K is non-Archimedean, we have that z ∈ ∞ . Moreover, by construction, x = z − R(z). Thus, T (x) = T (z − R(z)) = 0 for every x ∈ ∞ . We have reached a contradiction since the latter implies that T ≡ 0 on ∞ , but T c 0 = T ≡ 0.
We have proven the following result.
Theorem 5.8 If K is a non-spherically complete non-Archimedean field with non-trivial valuation, and c 0 and ∞ are defined over K, then the family of functionals on c 0 that cannot be extended to a functional on ( ∞ ) is c -lineable.
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