Denjoy-Carleman classes and lineability

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Denjoy-Carleman classes and lineability

Céline ESSER

Celine.Esser@ulg.ac.be

University of Liège – Institute of Mathematics

Workshop on Functional Analysis Valencia 2015 on the occasion of the 60th birthday of José Bonet

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Introduction

AC∞_function_f_{is analytic at}_x

0∈ Ωif its Taylor series atx0converges tofon an

open neighbourhood ofx0. Using Cauchy’s estimates, it is equivalent to have the

existence of a compact neighborhoodKofx0and of two constantsC, h > 0such that

sup

x∈K

Dkf (x)≤ Chkk! ∀k ∈ N0.

We say that a function isnowhere analytic on an intervalif it is not analytic at any point of the interval.

Many examples ofC∞_{nowhere analytic functions exist. An example was given by}

Cellérier (1890) with the function defined for all_{x ∈ R}by

f (x) = +∞ X n=0 sin(an_x) n!

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Introduction

sup

x∈K

f (x) = +∞ X n=0 sin(an_x) n!

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Introduction

sup

x∈K

f (x) = +∞ X n=0 sin(an_x) n!

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Introduction

Question.

How large is the set of nowhere analytic functions in the Fréchet spaceC∞_{([0, 1])}_?

Is it possible to constructlarge structuresof nowhere analytic functions?

Lineability

(Aron, Gurariy, Seoane-Sepúlveda 2005)

LetX be a topological vector space,M a subset ofX, andµa cardinal number.

(1) The setM islineableifM ∪ {0}contains an infinite dimensional vector subspace. If the dimension of this subspace isµ,M is said to beµ-lineable.

(2) When the above linear space can be chosen to be dense inX, we say thatM is (µ-)dense-lineable.

Results. Genericity (different notions) and extension using Gevrey classes • Morgenstern 1954

• Cater 1984

• Salzmann and Zeller 1955 • Bernal-Gonzalez 2008

• Bastin, E., Nicolay 2012

• Conejero, Jiménez-Rodríguez, Muñoz-Fernández and Seoane-Sepúlveda 2012

• Bartoszewicz, Bienias, Filipczak and Gła¸b 2013 • Bastin, Conejero, E. and Seoane 2014

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Introduction

Question.

Lineability

• Cater 1984

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Introduction

Question.

Lineability

• Cater 1984

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Denjoy-Carleman classes

Question. Similar results in the context of classes of ultradifferentiable functions?

An arbitrary sequence of positive real numbersM = (Mk)k∈N0 is called aweight

sequence.

Roumieu classes

LetΩbe an open subset ofRandM be a weight sequence. The spaceE{M }(Ω)is defined by

E{M }(Ω) :=f ∈ C∞(Ω) : ∀K ⊆ Ωcompact∃h > 0such thatkf kMK,h< +∞ ,

where kf kM K,h:= sup k∈N0 sup x∈K |Dk_{f (x)|} hk_M k .

Iff ∈ E_{{M }}(Ω), we say thatf isM-ultradifferentiable of Roumieu typeonΩ.

Particular case. The weight sequences(k!)_k∈N₀and((k!)α₎

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Denjoy-Carleman classes

sequence.

Roumieu classes

LetΩbe an open subset ofRandM be a weight sequence. The spaceE{M }(Ω)is defined by

Iff ∈ E_{{M }}(Ω), we say thatf isM-ultradifferentiable of Roumieu typeonΩ.

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Denjoy-Carleman classes

sequence.

Roumieu classes

LetΩbe an open subset of_RandM be a weight sequence. The spaceE{M }(Ω)is defined by

Iff ∈ E{M }(Ω), we say thatf isM-ultradifferentiable of Roumieu typeonΩ.

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Denjoy-Carleman classes

sequence.

Roumieu classes

LetΩbe an open subset of_RandM be a weight sequence. The spaceE{M }(Ω)is defined by

Iff ∈ E{M }(Ω), we say thatf isM-ultradifferentiable of Roumieu typeonΩ.

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Beurling classes

LetΩbe an open subset of_RandM be a weight sequence. The spaceE(M )(Ω)is defined by

E(M )(Ω) :=f ∈ C∞(Ω) : ∀K ⊆ Ωcompact, ∀h > 0, kf kMK,h< +∞ .

Iff ∈ E(M )(Ω), we say thatf isM-ultradifferentiable of Beurling typeonΩand we

use the representation

E(M )(Ω) = proj ←−−− K⊆Ω proj ←−− h>0 EM,h(K)

to endowE(M )(Ω)with a structure of Fréchet space.

Of course, we always haveE(M )(Ω) ⊆ E{M }(Ω).

Questions.

• When do we haveE{M }(Ω) ⊆ E(N )(Ω)?

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Beurling classes

LetΩbe an open subset of_RandM be a weight sequence. The spaceE(M )(Ω)is defined by

E(M )(Ω) :=f ∈ C∞(Ω) : ∀K ⊆ Ωcompact, ∀h > 0, kf kMK,h< +∞ .

Iff ∈ E(M )(Ω), we say thatf isM-ultradifferentiable of Beurling typeonΩand we

use the representation

E(M )(Ω) = proj ←−−− K⊆Ω proj ←−− h>0 EM,h(K)

to endowE(M )(Ω)with a structure of Fréchet space.

Of course, we always haveE(M )(Ω) ⊆ E{M }(Ω).

Questions.

• When do we haveE{M }(Ω) ⊆ E(N )(Ω)?

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General assumptions.

• We assume that any weight sequenceM is logarithmically convex, i.e.

M_k2≤ Mk−1Mk+1, ∀k ∈ N .

It implies that the spaceE{M }(Ω)is an algebra.

• We assume that any weight sequenceM is such thatM0= 1.

• We will often work with non-quasianalytic weight sequencesM, i.e. such that

+∞

X

k=1

(Mk)−1/k< +∞.

By Denjoy-Carleman theorem, it is equivalent to the fact that there exists non-zero functions with compact support inE{M }(R).

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M_k2≤ Mk−1Mk+1, ∀k ∈ N .

+∞

X

k=1

(Mk)−1/k< +∞.

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M_k2≤ Mk−1Mk+1, ∀k ∈ N .

+∞

X

k=1

(Mk)−1/k< +∞.

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Inclusions between Denjoy-Carleman classes

Notation. Given two weight sequencesM andN, we write

M_{N ⇐⇒} lim k→+∞ Mk Nk 1k = 0.

Proposition

LetM, N be two weight sequences and letΩbe an open subset of_R. Then

M_{N ⇐⇒ E}_{{M }}(Ω) ⊆ E(N )(Ω)

and in this case, the inclusion is strict.

Keys.

• IfM _N, then there exists a weight sequenceLsuch thatM_{L N}.

• The function θ(x) = +∞ X k=1 Mk 2k Mk−1 Mk k exp 2i Mk Mk−1 x

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Inclusions between Denjoy-Carleman classes

M_{N ⇐⇒} lim k→+∞ Mk Nk 1k = 0.

Proposition

M_{N ⇐⇒ E}_{{M }}(Ω) ⊆ E(N )(Ω)

Keys.

• IfM _N, then there exists a weight sequenceLsuch thatM_{L N}.

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Inclusions between Denjoy-Carleman classes

M_{N ⇐⇒} lim k→+∞ Mk Nk 1k = 0.

Proposition

M_{N ⇐⇒ E}_{{M }}(Ω) ⊆ E(N )(Ω)

Keys.

• IfM_N, then there exists a weight sequenceLsuch thatM_{L N}.

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Construction

Definition

We say that a function isnowhere inE{M }if its restriction to any open and non-empty subsetΩof_Rnever belongs toE{M }(Ω).

Proposition

Assume thatM andNare two weight sequences such thatM _N. IfM is non-quasianalytic, there exists a function ofE(N )(R)which is nowhere inE{M }.

Proof. From Lemma 1, there isN?_{such that}_M

N?

N.Applying recursively this lemma, we get a sequence(L(p)₎

p∈Nof weight sequences such that

M_L(1)_L(2)_{· · · L}(p)_{· · · N}?_N.

For every_{p ∈ N}, Lemma 2 allows us to consider a functionfp∈ E{L(p)_}(R)such that |Dj_f

p(0)| ≥ L (p)

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Construction

Definition

Proposition

Assume thatM andN are two weight sequences such thatM _N. IfM is non-quasianalytic, there exists a function ofE(N )(R)which is nowhere inE{M }.

N?

M_L(1)_L(2)_{· · · L}(p)_{· · · N}?_N.

For every_{p ∈ N}, Lemma 2 allows us to consider a functionfp∈ E{L(p)_}(R)such that |Dj_f

p(0)| ≥ L (p)

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Construction

Definition

Proposition

Assume thatM andN are two weight sequences such thatM _N. IfM is non-quasianalytic, there exists a function ofE(N )(R)which is nowhere inE{M }.

N?

M_L(1)_L(2)_{· · · L}(p)_{· · · N}?_N.

For every_{p ∈ N}, Lemma 2 allows us to consider a functionfp∈ E{L(p)_}(R)such that

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SinceM is non-quasianalytic,

∃φ ∈ E{M }(R), suppϕcompact, φ ≡ 1in a nbh of0.

Let{xp: p ∈ N0}be a dense subset ofRwithx0= 0. For everyp ∈ N, we can find

kp> 0such that the function

φp:= φ kp(· − xp)

has its support disjoint from{x0, . . . , xp−1}. We definegp

gp(x) := fp(x − xp) | {z } ∈E_{{L(p) }}(R) φp(x) | {z } ∈E{M }(R) ∈ E(N?)(R).

Letγp> 0be such that

sup

x∈R

|Dj_g

p(x)| ≤ γpNj?, ∀j ∈ N0

and define the functiongby

g := +∞ X p=1 1 γp2p gp.

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SinceM is non-quasianalytic,

∃φ ∈ E{M }(R), suppϕcompact, φ ≡ 1in a nbh of0.

Let{xp: p ∈ N0}be a dense subset ofRwithx0= 0. For everyp ∈ N, we can find

kp> 0such that the function

φp:= φ kp(· − xp)

has its support disjoint from{x0, . . . , xp−1}. We definegp

gp(x) := fp(x − xp) | {z } ∈E_{{L(p) }}(R) φp(x) | {z } ∈E{M }(R) ∈ E(N?)(R).

Letγp> 0be such that

sup

x∈R

|Dj_g

p(x)| ≤ γpNj?, ∀j ∈ N0

and define the functiongby

+∞

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1.g ∈ E(N )(R): for everyj ∈ N0and everyx ∈ R, we have +∞ X p=1 1 γp2p |Djgp(x)| ≤ +∞ X p=1 1 2pN ? j ≤ N ? j

which implies thatgbelongs toE{N?_}(R) ⊆ E_{(N )}(R).

2.gis nowhere inE{M }: By contradiction, assume that there exists an open subsetΩ

of_Rsuch thatg ∈ E{M }(Ω). Letp0∈ Nsuch thatxp0 ∈ Ω. Remark that

+∞ X p=p0 1 γp2p gp= g |{z} ∈E{M }(Ω)⊆E_{(L(p0 ) )}(Ω) − p0−1 X p=1 1 γp2p gp | {z } ∈E (L(p0 ) )(Ω) ∈ E(L(p0))(Ω).

But, since the support ofgpis disjoint ofxp0for everyp > p0, we also have

+∞ X p=p0 1 γp2p Djgp(xp0) = 1 γp02 p0 Djgp0(xp0) = 1 γp02 p0 Djfp0(0) ≥ 1 γp02 p0L (p0) j

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+∞ X p=p0 1 γp2p gp = g |{z} ∈E{M }(Ω)⊆E_{(L(p0 ) )}(Ω) − p0−1 X p=1 1 γp2p gp | {z } ∈E (L(p0 ) )(Ω) ∈ E(L(p0))(Ω).

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+∞ X p=p0 1 γp2p gp = g |{z} ∈E{M }(Ω)⊆E_{(L(p0 ) )}(Ω) − p0−1 X p=1 1 γp2p gp | {z } ∈E (L(p0 ) )(Ω) ∈ E(L(p0))(Ω).

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Remark. Given a sequence(L(p)₎

p∈Nsuch that

M _L(1)_L(2)_{· · · L}(p)_{· · · N}∗_N

and a dense subset{xp: p ∈ N0}ofR, we have constructed a functiongsuch that

• g ∈ E{N∗_}(R)

• ∈ E/ (L(p)₎(Ω)for every neighbourhoodΩofxp.

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Lineability

Proposition

Assume thatM _N. IfM is non quasianalytic, the set of functions ofE(N )(R)which

are nowhere inE{M }isc-lineable.

Idea of the proof. For everyt ∈ (0, 1), we set

L(t)_k := (Mk)1−t(Nk)t ∀k ∈ N0.

ThenM _L(t)_Nfor allt ∈ (0, 1)andL(t)_L(s)ift < s.

Remark that M _L(t2) L( 2t 3) L( 3t 4) · · · L(t) N, ∀t ∈ (0, 1).

and we can considergt∈ E{L(t)_}(R)which is not inE

L((1− 1p)t)(Ω)

, for any open

neighbourhoodΩofxpand for anyp ≥ 2. Then take

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Lineability

Proposition

Assume thatM _N. IfM is non quasianalytic, the set of functions ofE(N )(R)which

are nowhere inE{M }isc-lineable.

Idea of the proof. For everyt ∈ (0, 1), we set

L(t)_k := (Mk)1−t(Nk)t ∀k ∈ N0.

ThenM _L(t)_Nfor allt ∈ (0, 1)andL(t)_L(s)ift < s. Remark that M _L(2t) L( 2t 3) L( 3t 4) · · · L(t) N, ∀t ∈ (0, 1).

and we can considergt∈ E{L(t)_}(R)which is not inE

L((1− 1p)t)(Ω)

, for any open

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Dense-lineability

Remark. The set of polynomials is dense inE(N )(R). Let(tm)m∈Nbe a sequence of

different elements of(0, 1)and let(Ptm)m∈Nbe a dense sequence of polynomials in E(N )(R).

For every_{m ∈ N}, letkm> 0be such thatkmgtm ∈ Um, where{Um: m ∈ N}is a

0-neighbourhoods basis inE(N )(R).

We set

Dd= spanPt+ ktgt: t ∈ (0, 1)

wherekt= 1andPt= 0ift 6= tmfor everym ∈ N.

Proposition

Assume thatM _N. IfM is non quasianalytic, thenDdis dense inE(N )(R),

dim Dd= cand any non zero function ofDdis nowhere inE{M }. In particular, the set

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Dense-lineability

We set

Proposition

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Dense-lineability

We set

Proposition

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Dense-lineability

We set

Proposition

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Case of countable unions

LetN be a weight sequence and let(M(n))_n∈Nbe a sequence of weight sequences such thatM(n)

Nfor every_{n ∈ N}.

Question.

What about the set of functions ofE(N )(R)which are nowhere inS_n∈NE{M(n)_}?

Proposition

If there isn0∈ Nsuch that the weight sequenceM(n0)is non quasianalytic, the set of

functions ofE(N )(R)which are nowhere inS_n∈NE{M(n)_}isc-dense-lineable in E(N )(R).

Idea. Construct a weight sequenceP such that

[

n∈N

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Case of countable unions

LetN be a weight sequence and let(M(n))_n∈Nbe a sequence of weight sequences such thatM(n)

Nfor every_{n ∈ N}.

Question.

What about the set of functions ofE(N )(R)which are nowhere inS_n∈NE{M(n)_}?

Proposition

If there isn0∈ Nsuch that the weight sequenceM(n0)is non quasianalytic, the set of

functions ofE(N )(R)which are nowhere inS_n∈NE{M(n)_}isc-dense-lineable in E(N )(R).

Idea. Construct a weight sequenceP such that

[

n∈N

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Gevrey classes

They correspond to Roumieu classes given by the weight sequence

Mk := (k!)α, k ∈ N0.

Corollary

Letα > 1. The set of functions ofE((k!)α₎(R)which are nowhere inE_{(k!)β_}for every β ∈ (1, α), isc-dense-lineable inE((k!)α₎(R).

Proof. It suffices to apply the previous result the weight sequencesM(n)₍

n ∈ N) given by

M_k(n):= (k!)βn_, _{k ∈ N}

0,

where(βn)n∈Nis an increasing sequence of(1, α)that converges toα.

Proposition (Schmets, Valdivia 1991)

Letα > 1. The set of functions ofE((k!)α₎(R)which are nowhere inE_{(k!)β_}for every β ∈ (1, α)is residual inE((k!)α₎(R).

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Gevrey classes

Mk := (k!)α, k ∈ N0.

Corollary

Proof. It suffices to apply the previous result the weight sequencesM(n)(_{n ∈ N}) given by

M_k(n):= (k!)βn_,

k ∈ N0,

Proposition (Schmets, Valdivia 1991)

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Gevrey classes

Mk := (k!)α, k ∈ N0.

Corollary

Proof. It suffices to apply the previous result the weight sequencesM(n)(_{n ∈ N}) given by

M_k(n):= (k!)βn_,

k ∈ N0,

Proposition (Schmets, Valdivia 1991)

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More with weight functions

Definition

A functionω : [0, +∞) → [0, +∞)is called aweight functionif it is continuous, increasing and satisfiesω(0) = 0as well as the following conditions

(α) There existsL ≥ 1such thatω(2t) ≤ Lω(t) + L, t ≥ 0,

(β)

Z +∞

1

ω(t)

t2 dt < +∞,

(γ) log(t) = o(ω(t)))asttends to infinity, (δ) ϕω: t 7→ ω(et)is convex on[0, +∞).

The Young conjugate ofϕωis defined by

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Roumieu classes

Ifωis a weight function and ifΩis an open subset of_Rn, we define the space

E{ω}(Ω)ofω-ultradifferentiable functions of Roumieu typeonΩby

E{ω}(Ω) := {f ∈ E (Ω) : ∀K ⊂ Ωcompact∃m ∈ N such thatkf kK,m< +∞} ,

wherekf kK,m:= sup_α∈Nn 0 supx∈K|D α_{f (x)| exp −}1 mϕ ∗ ω(m|α|) < +∞.

Beurling classes

Ifωis a weight function and ifΩis an open subset of_Rn, the spaceE(ω)(Ω)of

ω-ultradifferentiable functions of Beurling typeonΩis defined by

E(ω)(Ω) := {f ∈ E (Ω) : ∀K ⊂ Ωcompact, ∀m ∈ N, pK,m(f ) < +∞} ,

where for every compact subsetKof_Rnand every_{m ∈ N}

pK,m(f ) := sup α∈Nn 0 sup x∈K |Dα_{f (x)| exp} −mϕ∗ ω |α| m .

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Roumieu classes

Ifωis a weight function and ifΩis an open subset of_Rn, we define the space

E{ω}(Ω)ofω-ultradifferentiable functions of Roumieu typeonΩby

E{ω}(Ω) := {f ∈ E (Ω) : ∀K ⊂ Ωcompact∃m ∈ N such thatkf kK,m< +∞} ,

wherekf kK,m:= sup_α∈Nn 0 supx∈K|D α_{f (x)| exp −}1 mϕ ∗ ω(m|α|) < +∞.

Beurling classes

Ifωis a weight function and ifΩis an open subset of_Rn, the spaceE(ω)(Ω)of

ω-ultradifferentiable functions of Beurling typeonΩis defined by

E(ω)(Ω) := {f ∈ E (Ω) : ∀K ⊂ Ωcompact, ∀m ∈ N, pK,m(f ) < +∞} ,

where for every compact subsetKof_Rnand every_{m ∈ N}

pK,m(f ) := sup α∈Nn 0 sup x∈K |Dα_{f (x)| exp} −mϕ∗ ω |α| m .

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Given two weight functionsωandσ, we write

ω_{σ ⇐⇒ σ(t) = o(ω(t))}ast → +∞.

Proposition

Letωandσbe two weight functions such thatω_σ. IfΩis a convex open subset of

Rn, thenE{ω}(Ω)is strictly included inE(σ)(Ω).

Proposition

Letωandσbe two weight functions such thatω_σ. The set of functions ofE(σ)(Rn)

which are nowhere inE{ω}is dense-lineable inE(σ)(Rn).

Idea.

• Existence: Baire category theorem.

• Lineability: Construct a sequence(ω(p)₎

p∈Nof weight functions such that

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Given two weight functionsωandσ, we write

ω_{σ ⇐⇒ σ(t) = o(ω(t))}ast → +∞.

Proposition

Letωandσbe two weight functions such thatω_σ. IfΩis a convex open subset of

Rn, thenE{ω}(Ω)is strictly included inE(σ)(Ω).

Proposition

Letωandσbe two weight functions such thatω_σ. The set of functions ofE(σ)(Rn)

which are nowhere inE{ω}is dense-lineable inE(σ)(Rn).

Idea.

• Existence: Baire category theorem.

• Lineability: Construct a sequence(ω(p)₎

p∈Nof weight functions such that

ω(1)

· · · ω(p)

ω(p+1)

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