ON THE CLOSED SUBSPACES OF UNIVERSAL SERIES IN BANACH SPACES AND FRÉCHET SPACES

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SPACES AND FRÉCHET SPACES

CHARPENTIER STÉPHANE

Abstract. We prove, in a general framework, the existence of a closed innite dimensional subspace consisting of universal series.

Introduction

The study of universal series began in 1914, when Fekete showed that there exists a Taylor series on [−1,1] whose subsequences of partial sums approximate uniformly any continuous function f on [−1,1]such that f(0) = 0. The main idea of this result is that whatever can be uniformly approximated by polynomials can also be approximated by subsequences of partial sums of a Taylor series. After this, a lot of examples of universal series appeared in dierent spaces: Menchov showed, for instance, the existence of many trigonometric series which approximate almost everywhere, along subsequences of its partial sums, any complex measurable 2π-periodic function on the torus. Other punctual universality results were obtained concerning Faber series, Laurent series, Jacobi series, harmonic expansions, etc. For such results, we refer to the two surveys [6] and [7].

It turns out that, when a universal series does exist, the set of universal series has some nice properties. This was developed in [5] (in a more general context) and later in [2]

and [14], where an abstract theory of universal series is exposed. Precisely, it is shown there that, as soon as there exists a universal series, then the set of universal series is both generic (it contains a denseG_δ set) and algebraically generic (it contains a dense subspace, except0.) This is done in a general framework which covers every known example.

Independently, Bayart studied in [1] the existence of a closed innite dimensional subspace of universal series. His work just deals with one particular example, that of universal Taylor series of holomorphic functions on the unit disk D. Our aim, in this paper, is to combine the ideas of [1] and [2] in order to show that, in the abstract framework of [2], we can always nd a closed innite dimensional subspace of universal series except0, as soon as one universal series does exist.

This topic is not new in the context of universality. Indeed, a similar study was done for the so-called universal sequences of operatorTn:X−→Y (see [4] and [12].) However, the sucient condition of [12] is not helpful here (see Remark 2.3 below.) It should be added that, in our context, no extra assumptions for the existence of a closed innite dimensional subspace of universal series is needed.

The paper is organized as follows. The rst part is devoted to the presentation of all concepts and results involved in this work, the second part presents the proof of the existence of a closed innite dimensional subspace, except0, of universal series in the most simple general case, namely when we deal with Banach spaces. In the third part, we partly generalize this result to the case of Fréchet spaces; in the fourth part, we are led to add some extra parameters, as the center of expansion of a universal series, in order to be able to manage most of the examples. In the last part, we give a brief glimpse of how our results can be illustrated by usual examples of universality.

2000 Mathematics Subject Classication. 30BXX, 30B30.

Key words and phrases. universal series.

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1. Framework and results

Let (X, d_X) be a metrizable vector space and let (x_n)_n, n ∈ N, be a sequence in X. We suppose in the whole paper that the topology of X is dened by a countable family of semi-norms denoted by(k.k_n)_n≥0 and therefore that dX is invariant by translation. Let alsoA⊂k^N,k=Ror C, be a Fréchet space and(ei)_i ⊂k^N be the sequence ink^Ndened by e_i = (0, . . . ,0,1,0, . . .) where 1 is at the i-th position. We denote by d_A the distance inA which makes it a Fréchet space and consequentlydA is invariant by translation. We assume that Averies the following conditions:

(1) A contains the polynomials ofk^N, that is to say the elements which can be written as a nite linear combination of the ei's, with coecients ink.

(2) The set of polynomials ink^N, denoted byG, is dense inA. (3) The coordinate projections

pi : A −→ k (a_n)_n 7−→ a_i are continuous for any i∈N.

We will need the notions of valuation and degree of a polynomial ofk^N:

Denition 1.1. The valuation of a polynomial P ⊆k^N is the index of its rst non-zero coecient, and its degree is the index of its last non-zero coecient; we denote them respectively byv(P) andd(P).

We dene the notion of unrestricted universality for an element of X, with respect to the sequence (xn)_n, as follows:

Denition 1.2. We say thata= (a_n)_n∈k^N is unrestrictively universal (with respect to (x_n)_n) if the family

PN

n=0a_nx_n

N is dense inX. We denote byU the set of unrestricted universal sequences.

Remark 1.1. Let us notice that ifU is non-empty, thenXis necessarily separable. We can even observe that there exists a dense subsequence inX of vectors of the formPN

n=0a_nx_n. These vectors are called polynomials of X with respect to (xn)_n.

This remark leads us to denote by the same letter a polynomial (a_n)_n in k^N and the polynomial P

nanxn in X, if there is no possible confusion. Otherwise, if a = (an)_n is more generally an element of A, we will denote by S_N(a) the partial sum of order N, PN

n=0a_ne_n in A, and we will distinguish it from S_N(a, X) = PN

n=0a_nx_n, the partial sum ofainX with respect to (xn)_n.

In this paper, we will be also interested in the notion of restricted universality:

Denition 1.3. An element a = (an)_n ∈ A is said to be restrictively universal (with respect to (x_n)_n) if, for every x ∈ X, there exists an increasing sequence (λ_N)_N∈

N of integers such that

(1) PλN

n=0anxnconverges to x whenN goes to innity, (2) PλN

n=0anen converges toawhen N goes to innity.

We denote byUA the set of restricted universal sequences.

Of course, every restricted universal sequence is unrestricted, and we notice that if(e_n)_n is a Schauder basis ofA, we have immediatelyU∩A=UA; so that the notion of restricted universal sequences has an interest when(en)_n is not a Schauder basis ofA.

Our result is based on the following one, given by [2], and which characterizes the non- emptiness of the setUA.

Theorem 1.1. Under the previous assumptions, the following are equivalent:

(1) U_A is non-empty.

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(2) For every ε > 0, for every x ∈ X, and for every integer p ≥ 0, there exists a polynomial (a_n)_n with valuation p such that

d_X X

n∈N

a_nx_n, x

!

< εand d_A X

n∈N

a_ne_n,0

!

< ε.

(3) U_A∪ {0} is a dense G_δ subset of A and it contains a dense subspace of A.

Our aim, in this paper, is to show that whenU_Ais non-empty, it contains automatically a closed innite dimensional subspace ofAexcept0. We will rst give a proof whenAis a Banach space; this proof will be partly based on the possibility to build a convenient basic sequence of the Banach space A. This trick fails when Ais a Fréchet space. Thus, in this latter case, we just obtain thatU∩A (and notU_A) contains a closed innite dimensional subspace ofAexcept0. However, in all concrete examples,U_Ais equal toU∩A. Therefore, let us say that the problem is solved in main cases.

2. Universality in Banach spaces

In this section, we deal with the case whereAis a Banach space, with a normk.k_A. For this, we will need to show how to build a normalized basic sequence in A whose elements can be chosen as polynomials with an arbitrary valuation. This is the purpose of the following lemma:

Lemma 2.1. Let ε >0 be a real number, and let(u0, . . . , un)⊆A be a nite family. For every integer v, there exists a polynomial u_n+1 ∈A such that v(u_n+1) ≥v, ku_n+1k_A = 1 and(2.1)

n

X

k=0

λ_ku_k A

≤(1 +ε)

n

X

k=0

λ_ku_k+λu_n+1 A

, for any scalars λand λ_k,0≤k≤n.

Remark 2.1. If (ε_n)_n≥0 is a sequence of positive real numbers such that Q∞

n=0(1 +ε_n) converges toK, by dening for example u0 = e0

ke₀k_A, then we can construct by induction a basic sequence(u_n)_n≥0 of constant of basicity not greater thanK, such that every term is built as in the previous lemma, by taking ε=ε_n at stepn.

Indeed, for every integerpandq withq≥p, Lemma 2.1 allows us to construct polynomials u_k,0≤k≤q, such that, for every scalars (λ_k)_0≤k≤q,

p

X

k=0

λ_ku_k A

≤

q−1

Y

k=p

(1 +ε_k)

q

X

k=0

λ_ku_k A

≤K

q

X

k=0

λ_ku_k A

.

Proof of Lemma 2.1. Letε >0be a real number and(u₀, . . . , u_n)⊆Abe a nite family of polynomials. Let alsov be an integer. We denote byEn+1 the nite dimensional subspace of Adened by:

E_n+1=span(e₀, . . . , e_v). Let(z_j)^l_j=0ⁿ⁺¹ be an ε

1 +ε-net of the unit sphere of span(u₀, . . . , u_n) and(ϕ_j)^l_j=0ⁿ⁺¹ be continuous linear functionals of norm1 over Asuch that, for every 0≤j≤ln+1,

ϕ_j(z_j) = 1.

As codim

∩^l_j=0ⁿ⁺¹ker (ϕ_j)

∩

∩^v_i=0ker (e^∗_i)

≤l_n+1+v+ 2, there exists an integerm_n+1 such that, if we denote

F_n+1=span e₀, . . . , e_m_n+1 , there exists u_n+1∈F_n+1 of norm1 such that

un+1 ∈

∩^l_j=0ⁿ⁺¹ker (ϕj)

∩

∩^v_i=0ker (e^∗_i)

.

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By construction, un+1 is a polynomial of valuation greater than v, of norm 1. It just remains to verify that u_n+1 satises inequality (2.1). Let λbe an arbitrary scalar and let j0 be one index such that

n

X

k=0

λ_ku_k

n

X

k=0

λ_ku_k A

−z_j₀ A

≤ ε 1 +ε. By denition, we have

1 = ϕ_j₀(z_j₀+λu_n+1)

≤ kz_j₀ +λu_n+1k_A

≤

z_j₀ −

n

X

k=0

λ_ku_k

n

X

k=0

λ_ku_k A

A

+

n

X

k=0

λ_ku_k+λ

n

X

k=0

λ_ku_k A

u_n+1

n

X

k=0

λ_ku_k A

A

≤ ε

1 +ε+

n+1

X

k=0

λ_ku_k

n

X

k=0

λ_ku_k A

A

, by noting λ_n+1=λ

n

X

k=0

λ_ku_k A

.

Hence

1− ε 1 +ε

n

X

k=0

λ_ku_k A

≤

n+1

X

k=0

λ_ku_k A

namely

n

X

k=0

λ_ku_k A

≤(1 +ε)

n+1

X

k=0

λ_ku_k A

.

Up to dividingλby

n

X

k=0

λ_ku_k A

, we obtain the result.

This lemma is just an improvement of the construction of a basic sequence in a Banach space, given by S. Mazur (see [10], Theorem 1.a.5, page 4). When Theorem 1.1 is the main tool to show that the setU contains a closed innite dimensional subspace ofA, this lemma allows us to replace U by U_A.

Theorem 2.1. Under the previous assumptions and notations, ifUA6=∅, then it contains a closed innite dimensional subspace of A, except 0.

Proof. We will build by induction simultaneously a family(f_n,k)_n≥0,_0≤k≤n of elements of Aand a convenient basic sequence(u_k)_k ofA, thanks to which we will dene a subspace as desired. For this purpose, let (Q_l)_l be a sequence of polynomials dense in X and let ϕ, ψ : N−→ N be two functions such that, given any (m, t)∈N², there exists an innite family (vk)_k ⊆ N such that (ϕ(vk), ψ(vk)) = (m, t) for every k. Let also (εn)_n≥0 be a sequence of positive real numbers such thatQ∞

n=0(1 +εn) converges to a real numberK. The order of the inductive building of the f_n,k's will be the following: rst, we dene f0,0 from u0, then we build f1,0 from f0,0, thenf1,1 fromu1, thenf2,0 from f1,0,f2,1 from f1,1 and so on... This order is important because we will take into account all the previous f_l,t for the construction of an f_n,k. Each f_n,k will consist of a rst part, g_n,k, which will approximate the polynomial Q_ϕ(n) in X and of a second part which will annihilate the

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value of g_n,k inA. This second part will be necessary to ensure convergence of (f_n,k)_n in A and to be sure that its limit is close tou_k.

At the beginning, we x u0 = e0

ke₀k_A and approximate Q_ϕ(0) in X by g0,0 = u0+P, the polynomial of k^N where P is given by the point (2) of Theorem 1.1, applied to the spaces (A, dA) and

X,k.k_ψ(0)

, with x = Q_ϕ(0) −u0, p = 2 and ε= 1

2⁴K. Next we put f0,0 = g0,0+Q, where Q is given by the point (2) of Theorem 1.1, for (A, dA) and

X,k.k_ψ(1)

, with x=−g_0,0, p=d(P) + 1and ε= 1

2⁴K. Then,g_0,0 and f_0,0 satisfy the following formulas:

g_0,0−Q_ϕ(0)

ψ(0) < 1 2⁴K kf_0,0k_ψ(1) < 1

2⁴K kf_0,0−u0k_A < 1

2³K.

Now, we deal with the inductive step. So, letn∈Nbe xed and suppose that polynomials f_j,l, for l ≤j ≤n−1 have been built. We denote by ≺ the lexicographical order onN². We will construct the polynomialsf_n,k following the lexicographical order. First, we build fn,k for k < nas follows.

We approximate the polynomialQ_ϕ(n)inXby settingg_n,k =fn−1,k+P, whereP is given by Theorem 1.1, applied to the spaces(A, dA) and

X,k.k_ψ(n)

, withx=Q_ϕ(n)−fn−1,k, p= max (d(f_j,l),(j, l)≺(n, k)) + 1andε= 1

2ⁿ⁺⁴K. Then, we putf_n,kbyf_n,k =g_n,k+Q, where Q is given by Theorem 1.1, for (A, d_A) and

X,k.k_ψ(n+1)

, with x = −g_n,k, p = d(P) + 1and ε= 1

2ⁿ⁺⁴K.

We nally build un and fn,n. We deduce un from un−1 by induction and fn,n from un. We put g_n,n = u_n +P, with u_n given by Lemma 2.1 with B = (u₁, . . . , un−1), v = pn = max (d(fj,l),(j, l)≺(n, n)) + 1 and ε=εn, and whereP comes from Theorem 1.1, applied to the spaces (A, d_A) and

X,k.k_ψ(n)

, where x=Q_ϕ(n)−un,p=d(un) + 1 and ε= 1

2ⁿ⁺⁴K. Notice that asunis a polynomial, the denition ofgn,nhas sense inX. Then we denef_n,n=g_n,n+Q, whereQis given by Theorem 1.1, for(A, d_A)and

X,k.k_ψ(n+1) with x = −g_n,n, p = d(P) + 1 and ε= 1

2ⁿ⁺⁴K. As above, for n ≥ k, g_n,k, u_k and f_n,k satisfy the following formulas

g_n,k−Q_ϕ(n)

ψ(n) < 1 2ⁿ⁺⁴K (2.2)

kf_n,kk_ψ(n+1) < 1 2ⁿ⁺⁴K (2.3)

kf_n+1,k−f_n,kk_A < 1 2ⁿ⁺⁴K (2.4)

kf_k,k−ukk_A < 1 2^k+3K (2.5)

and(u_k)_k is a basic sequence ofAwith constant of basicity not greater thanK, according to Remark 2.1.

Using inequality (2.4), we dene for every k∈Nan element f_k ofA by f_k=

∞

X

n=k

(f_n+1,k−f_n,k) +f_k,k = lim

n→∞f_n,k.

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By construction, the family(f_k)_k is linearly independent: indeed, as we have d(f_n₁_,k₁)< v(f_n₂_,k₂) for (n₁, k₁)≺(n₂, k₂)

and as the linear coordinate projectionspi (see the introduction) are supposed to be continuous, an identity like PN

k=0α_kf_k = 0 for an integer N can be obtained only if α_k = 0 for every k. Therefore, the subspaceE of A dened by

E = ( _∞

X

k=0

α_kf_k|

∞

X

k=0

α_kf_k converges )

is innite dimensional and we will verify that the closed innite dimensional subspace F = ¯E answers our question, i.e. that every element of F is universal. First of all, it is advisable to remark that, in fact, every element hof F can be written asP∞

k=0α_kf_k with α_k∈k, namely thatEis closed andF =E. Indeed, by construction and using inequalities (2.4) and (2.5), we have for anyk

kf_k−ukk_A =

∞

X

n=k

(fn+1,k−fn,k) +fk,k−uk

<

∞

X

n=k

1

2ⁿ⁺⁴K + 1 2^k+3K

< 1 2^k+2K. (2.6)

Moreover, as (uk)_k is a basic sequence ofAof constant of basicity not greater than K,u^∗_k is well-dened and ku^∗_kk_A∗ku_kk_A ≤ 2K; by construction, ku_kk_A = 1 for any k; therefore (2.6) gives us

∞

X

k=0

ku^∗_kk_A∗kf_k−ukk_A<

∞

X

k=0

1 2^k+1 = 1.

Then the Bessaga-Pelczynski criterion ensures that(f_k)_kis a basic sequence ofAequivalent to (uk)_k so thatE = ¯E=F.

Now we take an elementh=P∞

k=0α_kf_k∈E=F and we show thathis universal; given a polynomialQ_l of the enumeration of polynomials dense inA, it is sucient to prove that there exists a sequence Nj (depending onh andQ_l) of integers such that

S_N_j(h, X)−−−→

j→∞ Q_l for the topology of (X, d_X), and then it is sucient to prove that, for everyn∈N,

S_N_j(h, X)−Q_l

n−−−→

n→∞ 0.

We denote by k0 the smallest integer k such that α_k 6= 0 and by Nj the degree of the polynomial g_v_j_+k₀_,k₀. By multiplying each α_k by (α_k₀)⁻¹, we may suppose that α_k₀ = 1. By denition of ϕ and ψ, there exists an increasing sequence (v_j)_j such that

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(ϕ(vj+k0), ψ(vj+k0)) = (l, n) for every j. Then we have

SNj(h, X)−Q_l n =

SNj





vj+k0

X

k=k0

α_kf_k, X



−Q_l

ψ(vj+k0)

=

vj−1+k₀

X

k=k0

α_kf_v_j−1+k₀,k+g_v_j_+k₀_,k₀ −Q_l

ψ(vj+k0)

≤

vj−1+k0

X

k=k0

α_kf_v_j−1+k0,k

ψ(vj+k0)

+

g_v_j_+k₀_,k₀ −Q_l

ψ(vj+k0)

≤

vj−1+k₀

X

k=k0

αkfvj−1+k0,k

ψ(vj+k0)

+ 1

2^v^j^+k⁰⁺⁴K, using (2.2).

(2.7)

Moreover, using inequality (2.6), and the fact that ku_kk_A= 1 for anyk, we get

|α_k| ≤ 2K⁰ kf_kk_A, withkf_kk_A≥1− 1

2^k+3K⁰ for every k, whereK⁰ is the constant of basicity of(f_k)_k. Thus the sequence (α_k)_k is bounded, say byM; (2.7) entails that

S_N_j(h, X)−Q_l

n ≤ M

vj−1+k0

X

k=k0

f_v_j−1+k₀,k

ψ(vj+k0)+ 1 2^v^j^+k⁰⁺⁴K

≤ M(v_j−1)

2^v^j^+3+k⁰K + 1

2^v^j^+k⁰⁺⁴K, by (2.3)

−−−−→

j−→∞ 0as the sequence (vj)_j is increasing, which proves of the universality of h.

The proof of the theorem will be nished after verifying that SNj(h) converges toh in A. This easily comes from the construction of the f_k's by using the triangle inequality.

Indeed we have

h−SNj(h)

A ≤

X

k≥v_j+k0+1

αkfk

A

+

vj+k0

X

k=k0

αk



 X

n≥v_j+k0+1

(fn+1,k−fn,k)



 A

+ +

α_v_j_+k₀ f_v_j_+k₀_,v_j_+k₀ −g_v_j_+k₀_,v_j_+k₀ A

≤

X

k≥v_j+k0+1

α_kf_k A

+

vj+k0

X

k=k0

|α_k|



 X

n≥v_j+k0+1

1 2ⁿ⁺⁴K



+ +

α_v_j_+k₀

1 2^v^j^+k⁰⁺⁴K

≤

X

k≥v_j+k0+1

αkfk

A

+ 1

16K sup

k≥k₀

|α_k|

vj+k0

X

k=k0

1 2^v^j^+k⁰ +

+

α_v_j_+k₀

1 2^v^j^+k⁰⁺⁴K

−−−−→

j−→∞ 0,

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since the series X

k≥0

α_kf_k is convergent and the sequence(α_k)_k is bounded, which gives the

result.

Remark 2.2. Notice that we only use implication (1)⇒(2) of Theorem 1.1, which does not require the continuity of projectionspi's. In Theorem 1.1, this hypothesis is made to ensure that U_A is large; it is also the case in the last proof: continuity of the p_i's is crucial to show that the subspace E which we construct is innite dimensional.

The calculus which allows us to show that an element of E is universal is the main reason to assume thatX has a topology induced by a family of semi-norms. If this is not the case, our method is not successful in following this process, because we could not put the coecientsα_k out of the distance and use the fact that they are uniformly bounded.

Remark 2.3. More generally, let (T_n)_n be a sequence of operators from X to Y where X and Y are two Banach spaces. A vector x ∈ X is called universal for (Tn) if the set {T_nx;n∈N} is dense in Y. A sucient condition for the set of universal vectors to contain a closed innite dimensional subspace is given for instance in [12]. This condition requires that(Tn)satises the so-called Universality Criterion (see for instance [3].) In our context, the operator (T_N) is simply the application which maps (a_n) ∈A to the partial sum SN(a) = PN

n=0anxn in X. There is no reason why this sequence of maps should satisfy, in general, the Universality Criterion. Thus, the result of [12] does not cover our Theorem 2.1.

Theorem 2.1 can be almost generalized to the case of Fréchet spaces, that is the purpose of the forthcoming section.

3. Universality in Fréchet spaces

We keep the notations of the introduction, and for instanceAis now a Fréchet subspace ofk^N with an invariant by translation metricdA; we also suppose that there is a norm k.kc over (A, d_A)such that there exists a constant C >0satisfying

(3.1) kxk ≤d Cd_A(x,0),∀x∈A.

This condition, which means that A admits a continuous norm, is of course equivalent to the fact that the topology of A is generated by a family of norms. This assumption is currently done in similar contexts. For example, this is the case in Theorem 3.1 of [4]

where the authors give a sucient condition for some sets of universal vectors to contain a closed innite dimensional subspace (see Remark 2.3.) They exhibit a counterexample when this assumption is removed (Example 3.2, [4].) Moreover, Fréchet spaces of complex sequences which can be identied to spaces of holomorphic functions in one variable are typical spaces which satisfy (3.1).

In the present case, we do not have a result as good as the one obtained in Theorem 2.1 for Banach spaces: we will show that(A, d_A)contains a closed innite dimensional subspace whose non-zero elements are all universal (with respect toX), namely thatU∪{0}contains a closed innite dimensional subspace ofA; but it will fail to show that it is also the case of UA∪ {0}, except of course if we suppose that (en)_n is a basis of A, i.e. UA = U, which nevertheless cover most of the reasonable cases (as the one of spaces of holomorphic functions in one variable.) We state our result as follows:

Theorem 3.1. With the previous notations, ifU_A is non-empty, then U contains a closed innite dimensional subspace of A, except0.

Proof. For our purpose, the hypothesis (3.1) will permit us to treat Fréchet spaces almost as if they were Banach spaces in the point of view of basic sequence. In fact, we consider the Banach space

A,b k.kc

obtained by the completion ofAfor the norm k.kc which veries

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(3.1). The second part of the proof of Theorem 2 in the article [1] inspired ourself with this idea. Let(Q_l)_l≥1 be an enumeration of a dense family of polynomials ofX and letϕ,ψbe two functions fromNto Nsuch that for every couple of integers(l, t), there exist innitely many integers j such that (ϕ(j), ψ(j)) = (l, t). By using Theorem 1.1, we simultaneously construct, as in the proof of Theorem 2.1 and with the same notations, three families

(f_n,k)_n≥k

k≥0,

(g_n,k)_n≥k

k≥0 and (u_n)_n≥0 of polynomials of A (orX, with no possible confusions,) with convenient valuation and degree, such that

g_n,k−Q_ϕ(n)

ψ(n) < 1 2ⁿ⁺⁴K (3.2)

kf_n,kk_ψ(n+1) < 1 2ⁿ⁺⁴K (3.3)

kf_n+1,k\−fn,kk ≤CdA(fn+1,k, fn,k) < 1 2ⁿ⁺⁴K (3.4)

kf_k,k\−ukk ≤CdA(fk,k, uk) < 1 2ⁿ⁺³K, (3.5)

being in(A, dA)for the construction of

(f_n,k)_n≥k

k≥0and

(g_n,k)_n≥k

k≥0and in A,b k.kc for the construction of the basic sequence (u_n)_n≥1 of the latter space. It is important to notice that Theorem 1.1 is only applied to (A, d_A)and not to the completion of

A,b k.kc , even for the construction offk,k and to obtain inequality (3.5). Indeed, theun's are built in

A,b k.kc

as polynomials, and so they belong to(A, dA)so that we can use Theorem 1.1 to (A, d_A) in order to deduce the construction of f_k,k in(A, d_A) from u_k. Moreover, this process consisting in passing from(A, dA)to

A,b k.kc

is allowed by assumption (3.1). We then dene thef_k's in(A, d_A) by

f_k=X

n≥k

(f_n+1,k−f_n,k) +f_k,k

and we consider the closed innite dimensional subspace F of (A, d_A) dened by F = E¯^d^A

where E =





 X

k≥0

α_kf_k which converges in (A, d_A), α_k ∈k







and E¯^d^A is the closure of E in (A, dA). Notice that the convergence in (A, dA) implies the convergence in

A,b k.kc according to (3.1), hence F ⊆

A,b k.kc

. To nish the proof, it is sucient to verify that every element of F, seen as an element of

A,b k.kc

is universal (we are now working in X). For this, we consider the setEe =

nP

k≥0αkfk which converges in A,b k.kc

, αk∈k o and we notice that F ⊆E. Now, we obtain the universality of every element ofe Ee directly from hypothesis (3.1), inequalities (3.2), (3.3), (3.4) and (3.5), and the proof of Theorem 2.1, by using the fact that the sequence(un)_n is basic and normalized in the Banach space

A,b k.kc

and therefore that(fk)_k is also a basic sequence in A,b k.kc

.

Let us recall Theorem 30 of [2]:

Theorem (Theorem 30 of [2]). Under the previous assumptions, the following are equivalent:

(1) U ∩A is non-empty.

(2) U ∩A is a denseG_δ subset of A.

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(3) For everyε >0, for everyx∈X, there existsm, n∈N,m≥n, anda0, a1, . . . , am

in k such that d_X





n

X

j=0

a_jx_j, x



< ε and d_A





m

X

j=0

a_je_j,0



< ε.

Looking at this result, our Theorem 3.1 may be a little disappointing. Indeed, comparing our results in Banach spaces and in Fréchet spaces, we can wonder if assuming UA 6= ∅ in the second case is not too strong, since we just obtain that U ∩A contains a closed innite dimensional subspace. At this point, Theorem 30 of [2] can appear as the one to appeal in Fréchet case. So a natural question can be: can we replace hypothesis UA6=∅ byU ∩A6=∅in Theorem 3.1? However, for technical reasons (like the appearance of the two indices n and m that we cannot control,) it turns out that it is not so evident that we can apply Theorem 30 of [2] in order to positively answer the last question. Moreover, notice that we do not know if U ∩A contains a dense subspace of A under assumption U ∩A6=∅. In fact, it is not even clear that it contains a subspace of dimension 2.

4. Labeled universal series

In the theory of universal series, we may deal with the center of expansion of the series. For instance, in the context of Taylor or power series, we usually have to change the center of expansion, that is to say, we have to consider an application e_n, for any n ∈N, from a set L in k into the convenient set of functions, with en(ξ) = (z−ξ)ⁿ. In the previous sections, we did not take care about this, by considering two xed families in A and X, respectively (en)_n and (xn)_n and we just considered expansion with respect to them. Moreover, most of the cases need to work not only with a single space X, but with a countable union of such sets, for example when we work with entire functions that we approximate by universal series on dierent sets. This case is no more covered by our previous studies.

In [2], the authors extend their abstract theory of universal series to these labeled universal series. They showed that everything works in this latter context as well as if we consider only one center of expansion and a single set X. Essentially, they give a generalization of Theorem 1.1.

In this part, we will show that the existence of a closed innite dimensional subspace consisting in universal series remains true in these two more general frameworks, and even when we combine them to take into account both the centers of expansions and the countable union of Fréchet spaces. The generalization of Theorem 1.1 is still at the heart of the proof, which is totally similar to the ones of Theorem 2.1 and Theorem 3.1, but more technical. In fact, the more extra parameters we will deal with, the more technical and heavier will be the proof. That is the reasons why we will just explain what we have to add to the previous proofs, and how it works well.

4.1. The center of expansion. We keep the notations Gand X used previously. We will consider a set of universal series more general than the set A ⊆k^N in the three rst parts, in order to work not only in a space of real or complex sequences, but directly in a convenient general Fréchet space onk, where the notion of centers of expansion can have much sense. We denote by (E, d_E) this space equipped with its complete invariant by translation metric and we suppose that its topology is given by a family of norms. Let alsoL be a compact space, letξ₀ be a xed element ofL, and let us suppose that, for any n∈N, we have continuous maps

en : L−→E,xn : L−→X and Hn : L×E −→k^N, withH_n linear with respect to the second coordinate verifying

(4.1) H_n(ξ₀, e_m(ξ₀)) =δ_n,m, for every n, m∈N.

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The mapsenandxn generalize respectively the sequenceseninAandxn inX introduced in the rst part.

Denition 4.1. For any a ∈ G, we denote by ga=

∞

X

n=0

anen(ξ0) the polynomial in E associated to a. We dene the valuationv(ga) and the degree d(ga) by v(ga) =v(a) and d(g_a) =d(a), with the notations of the rst part.

We make the usual following assumptions:

(1) The set{g_a, a∈G} is dense in E.

(2) For every a ∈ G and every ξ ∈ L, the sets {n, H_n(ξ, g_a)6= 0} are nite and uniformly bounded with respect to ξ.

(3) For every a∈Gand every ξ∈L, (4.2)

∞

X

n=0

Hn(ξ, ga)en(ξ) =

∞

X

n=0

anen(ξ0) and the same is true in X

(4.3)

∞

X

n=0

H_n(ξ, g_a)x_n(ξ) =

∞

X

n=0

a_nx_n(ξ₀).

For example, we may consider E as the Fréchet space H(D) of holomorphic functions on the disk, equipped with the standard metric of uniform convergence on compact sets, X as the spaceA(K)of holomorphic functions onK continuous onK, whereK is a compact with connected complement and K∩D=∅,L as a compact subset ofDand

e_n(ξ) =x_n(ξ) = (z7→(z−ξ)ⁿ) andH_n(ξ, f) = f⁽ⁿ⁾(ξ) n!

for ξ∈Dand f ∈H(D). Previous assumptions are clearly veried in this case.

Note that, if E =A, if the mapsen (resp. xn) are constant equal toen(ξ0) = en ∈ A (resp. to x_n ∈ X) and H_n(ξ, a) = a_n for every ξ ∈ L and a = (a_n)_n ∈ A, then we are precisely in the case covered by the previous sections.

Denition 4.2. We say that an elementf ∈E is unrestrictively universal (in this framework) if, for everyx∈X, there exists a sequence(λ_n)_n of integers such that

sup

ξ∈L

d_X

λN

X

n=0

H_n(ξ, f)x_n(ξ), x

!

−−−−→

N→∞ 0.

We denote byUE,L the set of all unrestricted universal series.

The generalized notion of restricted universal series is dened as follows:

Denition 4.3. We say that an elementf ∈Eis restrictively universal (in this framework) if, for everyx∈X, there exists a sequence(λ_n)_n of integers such that

sup

ξ∈L

d_E

λN

X

n=0

Hn(ξ, f)en(ξ), f

!

−−−−→

N→∞ 0

(4.4)

sup

ξ∈L

d_X

λN

X

n=0

H_n(ξ, f)x_n(ξ), x

!

−−−−→

N→∞ 0.

(4.5)

We denote byU_E,L^R the set of all restricted universal series.

Theorem 1.1 remains true in this context (see [2], Theorem 2, page 8):

Theorem 4.1. Under the previous assumptions, the following are equivalent:

(1) U_E,L^R is non-empty.

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sup

ξ∈L

d_X X

n∈N

H_n(ξ, g_a)x_n(ξ), x

!

< ε and sup

ξ∈L

d_E X

n∈N

H_n(ξ, g_a)e_n(ξ),0

!

< ε.

d_X X

n∈N

a_nx_n(ξ₀), x

!

< ε andd_E X

n∈N

a_ne_n(ξ₀),0

!

< ε.

(4) U_E,L^R ∪ {0} contains a dense subspace ofE.

We are now able to give the generalization of Theorem 3.1 in the context of labeled universal series, taking into account the center of expansion:

Theorem 4.2. With the previous assumptions, if U_E,L^R is non empty, then U_E,L contains a closed innite dimensional subspace of E, except 0.

Idea of the proof. We keep the notations(Q_l)_l∈

Nand ϕ, ψ introduced in Theorem 3.1. By proceeding exactly as in the proof of the latter, by considering (en(ξ0))_n instead of (en)_n (resp. (x_n(ξ₀))_n instead of (x_n)_n) and by the assumptions (4.2) and (4.3), we construct three families (gn,k)_n≥k≥0, (fn,k)_n≥k≥0 and (un)_n≥0 of polynomials in E, (or X with no possible confusions,) with convenient valuation and degree as in the proof of Theorem 3.1, such that

sup

ξ∈L

∞

X

j=0

Hj(ξ, g_n,k)xj(ξ)−Q_ϕ(n) ψ(n)

< 1 2^N+4K (4.6)

sup

ξ∈L

∞

X

j=0

Hj(ξ, fn,k)xj(ξ) ψ(n)

< 1 2ⁿ⁺⁴K (4.7)

dE(f_n+1,k, f_n,k) < 1 2ⁿ⁺⁴K (4.8)

kf_k,k\−u_kk < 1 2ⁿ⁺³K, (4.9)

and (un)_n≥0 being a basic sequence in

E,b k.kc

, the completed space ofE for k.k, one of the norms dening the topology of E. Then, we construct a family (f_k)_k≥0 in E, using inequality (4.8) by

f_k=X

n≥k

(f_n+1,k−f_n,k) +f_k,k for every k∈N, and we dene the closed subspaceF of E

F =





 X

k≥0

α_kf_k which converges in (E, d_E),α_k ∈k







dE

.

F is innite dimensional because of the assumption (4.1) and the construction of the f_k's. The proof of the universality of every element ofF is quite similar to the one used for Theorem 3.1 and works well after noticing that the sup is not an obstacle for the

calculus. We do not go into details.

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4.2. Universality in a countable family of Fréchet spaces. We keep the notations G, E,L, e_n,H_n and ξ₀ of the previous part. In the proof of Theorem 2.1, Theorem 3.1, or Theorem 4.2, the construction of the family (fk)_k≥0, which is at the core of them, is such that certain parameters were taken into account. Namely, in the latter, the procedure was to approximate, for a countable family of norms, a countable family of polynomials simultaneously by some countable convenient families of other polynomials. For this, the strategy is to build a sequence of polynomials with multiple index; each of these index should correspond to an extra parameter through the functionsϕ andψ. More generally, this method allows us to add some other countable extra parameters, up to get the formalism more complicated.

Therefore, we can extend Theorem 4.2 to the case of a countable family of Fréchet spaces ((Xn, dn))_n≥0 (where dn is the metric ofXn) instead of a single one X, and we even may work with a spaceLwhich is not compact, but which is an union of an exhaustive sequence (Lm)_m of compact subsets of it, such that ξ0 ∈ L0. Let also (x_n,k)_k≥0 be a sequence of continuous maps fromL to X_n, for any n∈N. We shall modify the assumptions (1), (2) and (3) of the previous part by the following:

(1) The set{g_a, a∈G} is dense in E.

(2) For every a ∈ G and every ξ ∈ L, the sets {n, H_n(ξ, ga)6= 0} are nite and uniformly bounded with respect to ξ∈L_m, for anym∈N.

(3) For every a∈G, every ξ∈L, (4.10)

∞

X

n=0

Hn(ξ, ga)en(ξ) =

∞

X

n=0

anen(ξ0) (4) For every a∈G, every ξ∈Land every n≥0,

(4.11)

∞

X

k=0

H_k(ξ, ga)x_n,k(ξ) =

∞

X

k=0

a_kx_k(ξ0).

In this context, we redene the notion of unrestricted universal series as follows:

Denition 4.4. We say that an elementf ∈E is unrestrictively universal (in this framework) if, for everyn∈Nand every x∈Xn, there exists a sequence(λn)_n of integers such that, for every m≥0,

sup

ξ∈Lm

d_X

λN

X

k=0

H_k(ξ, f)x_n,k(ξ), x

!

−−−−→

N→∞ 0.

We denote byUE,L the set of all unrestricted universal series.

And then, the notion of restricted universal series:

Denition 4.5. We say that an elementf ∈Eis restrictively universal (in this framework) if, for every n∈Nand every x ∈X_n, there exists a sequence(λ_n)_n of integers such that, for every m≥0,

sup

ξ∈Lm

dE λN

X

k=0

Hk(ξ, f)ek(ξ), f

!

−−−−→

N→∞ 0

(4.12)

sup

ξ∈Lm

dX λN

X

n=0

Hk(ξ, f)xn,k(ξ), x

!

−−−−→

N→∞ 0.

(4.13)

We denote byU_E,L^R the set of all restricted universal series.

As Theorem 4.1 also extends to this framework (Theorem 3, [2]), it suces to consider four functions ϕ, ψ, ν, κfrom N to N such that, for every (l, m, t, i) ∈ N⁴, there exists a increasing sequence(vk)_k such that(ϕ(vk), ψ(vk), ν(vk), κ(vk)) = (l, m, t, i) for everyk, and to follow the same proof as usual. Then, in this more general framework, we also have:

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Theorem 4.3. With the previous assumptions, if U_E,L^R is non empty, then UE,L contains a closed innite dimensional subspace of E, except 0.

5. Applications

This part is devoted to a short illustration of the previous results in some dierent frameworks, through classical examples of universality.

5.1. Applications in Banach spaces. We may rst illustrate Theorem 2.1 by consider- ingAto be the Banach spacesc₀ orl^p,1≤p <∞; with their natural topologies, they are the more standard Banach subspaces of C^N verifying hypothesis 1 to 3 of Section 1, and for which the canonical sequence (en)_n is basic. In [2], the authors apply their abstract theory to get the following:

Proposition 5.1. LetA=c₀orA=l^p,1≤p <∞, endowed with their natural topologies.

Then there exists a sequence a= (an)_n in A which veries the following property:

For every compact set K ⊆ C, K ∩D = ∅, with connected complement, for every function h∈A(K), there exists a sequence (λn)_n in Nsuch that

λn

X

j=0

a_jz^j−h(z) ∞,K

−−−→n→∞ 0

where k.k_∞,K is the supremum norm on K. Moreover, if we denote by U_A the set of such sequencesa, thenU_A is a denseG_δ subset of A, and it contains a dense vector subspace of A, except 0.

Let us also notice that, combining results in Section 1 and Section 4, it is clear that working with a Banach space A on one side and a countable family of Fréchet spaces (Xn, dn) on the other side yields the same conclusion than Theorem 2.1 under the same usual assumptions. Then, with the previous notations, let(X_n, d_n)be the countable family

A(K_n),k.k_∞,n

whereA(K_n) is the set of continuous functions onK_n, holomorphic on

◦

K_n, and where (K_n)_n is a family of compact sets ofC, withK_n∩D=∅,K_n^c connected, and such that every compact set K ⊆C, withK∩D=∅, K^cconnected, is contained in some K_n. Thanks to such a sequence (K_n)_n, it is clear that an application of Proposition 5.1 and of Theorem 4.3 in the easier present context gives the following result:

Proposition 5.2. With the notations of Proposition 5.1, the set UA contains a closed innite dimensional subspace of A, except0.

As in [2], we shall state a similar result in the context of Hardy or Bergman spaces, i.e whenA=H^p(D)or A=A^p(D),1≤p <∞. Indeed, as before, Corollary 8 in [2] ensures that we have the following:

Proposition 5.3. LetA=H^p(D) orA=A^p(D), 1≤p <∞. Then there exists a closed innite dimensional subspace F of A, except0, consisting of universal Taylor series, in the sense that for every f =P

nanzⁿ∈F, for every compact setK⊆C, K∩D=∅ with K^c connected, for every h∈A(K), there exists a sequence(λ_n)_n⊆N such that

λn

X

n=0

a_nzⁿ−h(z) _∞,K

−−−→n→∞ 0, where k.k_∞,K is the supremum norm on K.

A classical example of universal series is that of universal trigonometric series of Mencho in the periodic case; see [11] [8] and [2].

The dierence with our assumptions in Section 1 is that the topology of X is not induced by a countable family of seminorms. X is the space of complex measurable functions on

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T={z∈C: |z|= 1}modulo the equivalence relation identifying two functions if they are identical o a subset of Twith zero Lebesgue measure. If f, g∈X then

p_X(f, g) = Z

T

|f −g|

1 +|f −g|dλ

whereλ is the Lebesgue measure onT. However one can easily check that all our results hold in this case with A = c0 or A = l^p, p > 2. The universal series obtained in this way realize approximations in measure; but it turns out that these universal trigonometric series are identical with the universal trigonometric series realizing approximations in the almost everywhere sense. Thus we obtain the following:

Proposition 5.4. Let A=c₀ or l^p, p >2. There exists a sequence a= (a_n)_n≥0∈A such that, for every2π-periodic complex measurable functionh:R→C, there exists a sequence (λ_n)_n>1 ⊆Nso thatPλn

j=0a_je^ijx converges toh(x) almost everywhere asngoes to innity.

The set of such sequencesa∈Acontains a closed innite dimensional subspace ofAexcept 0.

We can also have an analogous result in the torus T^N (see [2].)

Now, letp≥0. If for everyp > p₀ there exists a universal series inl^p, then there exists a universal series in∩_p>p₀l^p. This follows from the Abstract Theory ([2]) in combination with the denition of the topology in the Fréchet space ∩_p>p₀l^p and is established in [15] (As a preprint, this can be found in the technical reports series of the Department of Mathematics and Statistics of the University of Cyprus.) This fact, forp0 = 2, in combination with the fact that there exists a universal trigonometric series in eachl^p,p >2, implies the existence of a universal trigonometric series in ∩_p>p₀l^p. The result of Section 3 implies now that

∩_p>p₀l^p contains a closed innite dimensional subspace whose non-zero elements dene universal trigonometric series of analytic type.

5.2. Applications in Fréchet spaces. In [9], the authors show the existence of universal series in A=∩_p>1l^p with respect to some sequences (x_n)_n in abstract Fréchet spaces X, up to a special condition on(xn)_n. More precisely, except the usual assumptions made in the previous sections, they introduce the following condition, which they call (D):

Condition (D). For every nite setI ⊂N, there exists distinct indicesj_n(i),n∈N,i∈I, such thatx_j_n_(i) −−−→

n→∞ xi.

We suppose thatA=∩_p>1l^pis endowed with its usual topology of Fréchet space induced by a family of norms. The main abstract theorem they obtain is stated as follows:

Theorem 5.1. With the current notations, assume that (x_n)_n satises Condition (D) and that the set of all nite linear combinations of the xn's is dense inX. Then U∩p>1l^p 6=∅.

Moreover, U∩_p>1l^p is a dense G_δ subset of A = ∩_p>1l^p and it contains a dense vector subspace of A, except0.

Let notice that the assumption that the set of all nite linear combinations of thexn's is dense in X is not a restriction for us, since it is a consequence of the fact that U_A is non-empty, what we have to assume in all our results in Section 2 and Section 3. Theorem 5.1 just reveals that, in this context, we can replace hypothesisU_A6=∅by Condition (D) and span(x_n, n∈N) =X. With this in mind, Theorem 5.1 and Theorem 4.3 of Section 3 yields this partially abstract result:

Theorem 5.2. With the usual notations, set A = ∩_p>1l^p and assume that (xn)_n ⊆ X satises Condition (D) and that span(xn, n∈N) = X. Then, U ∩A = UA contains a closed innite dimensional subspace, except 0.

Remark 5.3. We can easily see that Theorem 5.1 and Theorem 5.2 remain true if we replace

∩_p>1l^p by any l^p with 1 < p < ∞ or c0. Therefore, it may be seen as generalization of examples given in Section 5.1, except for the caseA=l¹; indeed the authors of [9] give an example where the statement of Theorem 5.1 does not hold whenA=l¹.