On the mutually non isomorphic ℓp(ℓq) spaces

Cembranos, Pilar; Mendoza, José

doi:10.1002/mana.201010056

Cited by 9 publications

(9 citation statements)

References 11 publications

(9 reference statements)

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“…This corresponds with the locally convex case. By [, Theorem 1.1] the spaces

ℓ_{p_{0}} (ℓ_{q_{0}})

and

ℓ_{p_{1}} (ℓ_{q_{1}})

are isomorphic if and only if

p_{0} = p_{1}

and

q_{0} = q_{1}

. Case 2 :

r_{0} \neq r_{1}

and we are not in Case 1. Without loss of generality,

r_{1} < r_{0}

so that

r_{1} < 1

.…”

Section: Preparatory Results and Main Theoremmentioning

confidence: 99%

“…This paper goes hand by hand with the aforementioned article , which inspired our work and where the reader will find a complete introduction as well as the necessary background to the problem we address.…”

Section: Introductionmentioning

confidence: 99%

“…To fix the specific notation that is not covered in , let us recall that a quasi‐normed space X is a locally bounded topological vector space. This is equivalent to saying that the topology on X is induced by a quasi‐norm , i.e., a map

∥ \cdot ∥ : X \to [0, \infty)

with the following properties:

∥ x ∥ = 0

if and only if

x = 0

∥ α x ∥ = | α | ∥ x ∥

α \in R, x \in X

, there is a constant

κ \geq 1

so that for any x and y in X we have

∥ x + y ∥ \leq κ (∥ x ∥ + ∥ y ∥) .

…”

Section: Introductionmentioning

confidence: 99%

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On the mutually non isomorphic ℓp(ℓq) spaces, II

Albiac

Ansorena

2014

Math. Nachr.

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This note is a companion to the article On the mutually non isomorphic ℓp(ℓq) spaces published in this journal, in which P. Cembranos and J. Mendoza showed that {ℓp(ℓq):1≤p,q≤∞} is a collection of mutually non isomorphic Banach spaces [5]. We now complete the picture by allowing the non‐locally convex relatives to be part of their natural family and see that, in fact, no two members of the extended class {ℓp(ℓq):0

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“…This corresponds with the locally convex case. By [, Theorem 1.1] the spaces

ℓ_{p_{0}} (ℓ_{q_{0}})

and

ℓ_{p_{1}} (ℓ_{q_{1}})

are isomorphic if and only if

p_{0} = p_{1}

and

q_{0} = q_{1}

. Case 2 :

r_{0} \neq r_{1}

and we are not in Case 1. Without loss of generality,

r_{1} < r_{0}

so that

r_{1} < 1

.…”

Section: Preparatory Results and Main Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

∥ \cdot ∥ : X \to [0, \infty)

with the following properties:

∥ x ∥ = 0

if and only if

x = 0

∥ α x ∥ = | α | ∥ x ∥

α \in R, x \in X

, there is a constant

κ \geq 1

so that for any x and y in X we have

∥ x + y ∥ \leq κ (∥ x ∥ + ∥ y ∥) .

…”

Section: Introductionmentioning

confidence: 99%

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On the mutually non isomorphic ℓp(ℓq) spaces, II

Albiac

Ansorena

2014

Math. Nachr.

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“…We have shown in [3] the result for p 0 , p 1 , q 0 , q 1 ∈ [1, +∞], therefore, we only have to consider the cases in which at least one of the indices is equal to 0. Of course, it is enough to consider the cases p 0 = 0 and q 0 = 0.…”

mentioning

confidence: 96%

“…In [6, footnote of p. 242] Triebel says he has learnt from Pełczyński that the p ( q ) spaces also make a family of mutually nonisomorphic spaces (see also [3]). It is natural to ask if this is still true if we add to this family the spaces c 0 ( p ) and p (c 0 ).…”

mentioning

confidence: 98%

The Banach spaces ℓ∞(c0) and c0(
Cembranos
¹
,
Mendoza
²

2010
Journal of Mathematical Analysis and Applications
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The statement of the title is proved. It follows from this that the spaces c 0 ( p ), p (c 0 ) and p ( q ), 1 p, q +∞, make a family of mutually non-isomorphic Banach spaces.In this note we use standard notation in Banach space theory as in [1] or [5].Some months ago Félix Cabello asked us: are the Banach spaces ∞ (c 0 ) and c 0 ( ∞ ) isomorphic? In this note we show that the answer is "no". In fact, we will prove that ∞ (c 0 ) is not even isomorphic to a quotient of c 0 ( ∞ ). We will also get some consequences of this result. To prove it we need a preliminary lemma.Lemma 1. Let W be a relatively weakly compact subset of c 0 and 0 < λ < 1, then there exists a norm one vector y ∈ c 0 such that

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Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]^d

Albiac

Ansorena

2016

Math. Nachr.

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The aim of this paper is to establish the isomorphic classification of Besov spaces over [0, 1]d. Using the identification of the Besov space Bp,qαfalse(false[0,1false]dfalse) with the ℓq‐infinite direct sum false(⊕n=1∞ℓpnfalse)q of finite‐dimensional spaces ℓpn (which holds independently of the dimension d≥1 and of the smoothness degree of the space α>0) we show that Bp,qαfalse(false[0,1false]dfalse), 0

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On the mutually non isomorphic ℓp(ℓq) spaces

Abstract: We extend a result of Pełczyński showing that { p ( q ) : 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of p ( q ) are also given.

Cited by 9 publications

References 11 publications

On the mutually non isomorphic ℓp(ℓq) spaces, II

On the mutually non isomorphic ℓp(ℓq) spaces, II

The Banach spaces ℓ∞(c0) and c0(
Cembranos
¹
,
Mendoza
²

2010
Journal of Mathematical Analysis and Applications
Self Cite
6
0
0
0
View full text Add to dashboard Cite

Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]^d

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On the mutually non isomorphic ℓp(ℓq) spaces

Abstract: We extend a result of Pełczyński showing that { p ( q ) : 1 ≤ p, q ≤ ∞} is a family of mutually non isomorphic Banach spaces. Some results on complemented subspaces of p ( q ) are also given.

Cited by 9 publications

References 11 publications

On the mutually non isomorphic ℓp(ℓq) spaces, II

On the mutually non isomorphic ℓp(ℓq) spaces, II

The Banach spaces ℓ∞(c0) and c0(Cembranos1, Mendoza2 2010Journal of Mathematical Analysis and ApplicationsSelf Cite6000View full textAdd to dashboardCite

Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]d

Contact Info

Product

Resources

About

The Banach spaces ℓ∞(c0) and c0(
Cembranos
¹
,
Mendoza
²

2010
Journal of Mathematical Analysis and Applications
Self Cite
6
0
0
0
View full text Add to dashboard Cite

Isomorphic classification of mixed sequence spaces and of Besov spaces over [0, 1]^d