![Page 1: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/1.jpg)
Atomic decompositions in locally convex spaces
Jornadas de Analisis Matematico
Alicante
Juan Miguel Ribera Puchades
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 1 / 22
![Page 2: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/2.jpg)
Aim
Aim
Aim
Our aim is discuss atomic decompositions on no normable Frechet Spaces.
Atomic Decompositions have been investigated by Carando, Casazza,Grochenig, Lassalle, Korobeınik, Schmidberg, Taskinen and others.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 2 / 22
![Page 3: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/3.jpg)
Aim
Aim
Aim
Our aim is discuss atomic decompositions on no normable Frechet Spaces.
Atomic Decompositions have been investigated by Carando, Casazza,Grochenig, Lassalle, Korobeınik, Schmidberg, Taskinen and others.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 2 / 22
![Page 4: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/4.jpg)
Introduction
Outline
1 Introduction
2 Atomic Decompositions
3 Duality
4 Unconditional atomic decompostions
5 Example
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 3 / 22
![Page 5: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/5.jpg)
Introduction
Definition
Let E be a Hausdorff locally convex space.
Definition
Let {xj}∞j=1 ⊂ E and let {x ′j}∞j=1 ⊂ E ′, we say that({x ′j}, {xj}
)is an
atomic decomposition of E if
x =∞∑j=1
x ′j (x) xj , for all x ∈ E ,
the series converging in E .
We denote by ω the space KN endowed by the product topology.A sequence space K(N) ⊂
∧⊂ KN is a lcs such that there exists a
continuous inclusion in ω.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 4 / 22
![Page 6: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/6.jpg)
Introduction
Definition
Let E be a Hausdorff locally convex space.
Definition
Let {xj}∞j=1 ⊂ E and let {x ′j}∞j=1 ⊂ E ′, we say that({x ′j}, {xj}
)is an
atomic decomposition of E if
x =∞∑j=1
x ′j (x) xj , for all x ∈ E ,
the series converging in E .
We denote by ω the space KN endowed by the product topology.A sequence space K(N) ⊂
∧⊂ KN is a lcs such that there exists a
continuous inclusion in ω.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 4 / 22
![Page 7: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/7.jpg)
Introduction
Definition
Let E be a Hausdorff locally convex space.
Definition
Let {xj}∞j=1 ⊂ E and let {x ′j}∞j=1 ⊂ E ′, we say that({x ′j}, {xj}
)is an
atomic decomposition of E if
x =∞∑j=1
x ′j (x) xj , for all x ∈ E ,
the series converging in E .
We denote by ω the space KN endowed by the product topology.
A sequence space K(N) ⊂∧⊂ KN is a lcs such that there exists a
continuous inclusion in ω.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 4 / 22
![Page 8: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/8.jpg)
Introduction
Definition
Let E be a Hausdorff locally convex space.
Definition
Let {xj}∞j=1 ⊂ E and let {x ′j}∞j=1 ⊂ E ′, we say that({x ′j}, {xj}
)is an
atomic decomposition of E if
x =∞∑j=1
x ′j (x) xj , for all x ∈ E ,
the series converging in E .
We denote by ω the space KN endowed by the product topology.A sequence space K(N) ⊂
∧⊂ KN is a lcs such that there exists a
continuous inclusion in ω.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 4 / 22
![Page 9: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/9.jpg)
Introduction
Examples
Examples
Let {ei}2i=1 be an orthonormal basis of a two-dimensional vectorspace V with inner product, then
x1 = e1, x2 = e1 − e2, x3 = e1 + e2
x ′1 =1
3e1, x
′2 =
1
3e1 −
1
2e2, x
′3 =
1
3e1 +
1
2e2
is an atomic decomposition of V .
Let {ei}∞i=1 an orthonormal basis of Hilbert space H, then
{xj}∞j=1 = {e1, e1, e2, e3, . . .}
{x ′j}∞j=1 =
{1
2e1,
1
2e1, e2, e3, . . .
}is an atomic decomposition of H.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 5 / 22
![Page 10: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/10.jpg)
Introduction
Examples
Examples
Let {ei}2i=1 be an orthonormal basis of a two-dimensional vectorspace V with inner product, then
x1 = e1, x2 = e1 − e2, x3 = e1 + e2
x ′1 =1
3e1, x
′2 =
1
3e1 −
1
2e2, x
′3 =
1
3e1 +
1
2e2
is an atomic decomposition of V .
Let {ei}∞i=1 an orthonormal basis of Hilbert space H, then
{xj}∞j=1 = {e1, e1, e2, e3, . . .}
{x ′j}∞j=1 =
{1
2e1,
1
2e1, e2, e3, . . .
}is an atomic decomposition of H.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 5 / 22
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Introduction
Examples
Examples
Let E be a lcs with a Schauder basis {ej}∞j=1 ⊂ E , the coefficient
functionals will be denoted by {e ′j}∞j=1 ⊂ E ′. If({e ′j}, {ej}
)is a
Schauder basis in a lcs E , then it is an atomic decomposition for E .
Let E be a lcs and let P : E → E be a continuous linear projection. If({x ′j}, {xj}
)is an atomic decomposition for E , then(
{P ′(
x ′j
)}, {P (xj)}
)is an atomic decomposition for P (E ).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 6 / 22
![Page 12: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/12.jpg)
Introduction
Examples
Examples
Let E be a lcs with a Schauder basis {ej}∞j=1 ⊂ E , the coefficient
functionals will be denoted by {e ′j}∞j=1 ⊂ E ′. If({e ′j}, {ej}
)is a
Schauder basis in a lcs E , then it is an atomic decomposition for E .
Let E be a lcs and let P : E → E be a continuous linear projection. If({x ′j}, {xj}
)is an atomic decomposition for E , then(
{P ′(
x ′j
)}, {P (xj)}
)is an atomic decomposition for P (E ).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 6 / 22
![Page 13: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/13.jpg)
Atomic Decompositions
Outline
1 Introduction
2 Atomic Decompositions
3 Duality
4 Unconditional atomic decompostions
5 Example
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 7 / 22
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Atomic Decompositions
Barrelled locally convex spaces are those satisfying the uniformboundedness principle (Banach-Steinhaus’ Theorem).
Example
Frechet spaces (complete and metrizable locally convex spaces).
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 8 / 22
![Page 15: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/15.jpg)
Atomic Decompositions
Barrelled locally convex spaces are those satisfying the uniformboundedness principle (Banach-Steinhaus’ Theorem).
Example
Frechet spaces (complete and metrizable locally convex spaces).
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 8 / 22
![Page 16: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/16.jpg)
Atomic Decompositions
Barrelled locally convex spaces are those satisfying the uniformboundedness principle (Banach-Steinhaus’ Theorem).
Example
Frechet spaces (complete and metrizable locally convex spaces).
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 8 / 22
![Page 17: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/17.jpg)
Atomic Decompositions
Barrelled locally convex spaces are those satisfying the uniformboundedness principle (Banach-Steinhaus’ Theorem).
Example
Frechet spaces (complete and metrizable locally convex spaces).
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 8 / 22
![Page 18: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/18.jpg)
Atomic Decompositions
Space structure
Let {xj} ⊂ E be a sequence of non-zero elements.
∧:=
α = (αj)j ∈ ω :∞∑j=1
αjxj is convergent in E
⊂ ω,
Q =
qp
((αj)j
):= sup
np
n∑j=1
αjxj
, for all p ∈ P
.
Theorem
If E is a complete lcs then (∧,Q) is a complete lcs.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 9 / 22
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Atomic Decompositions
Space structure
Let {xj} ⊂ E be a sequence of non-zero elements.
∧:=
α = (αj)j ∈ ω :∞∑j=1
αjxj is convergent in E
⊂ ω,Q =
qp
((αj)j
):= sup
np
n∑j=1
αjxj
, for all p ∈ P
.
Theorem
If E is a complete lcs then (∧,Q) is a complete lcs.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 9 / 22
![Page 20: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/20.jpg)
Atomic Decompositions
Space structure
Let {xj} ⊂ E be a sequence of non-zero elements.
∧:=
α = (αj)j ∈ ω :∞∑j=1
αjxj is convergent in E
⊂ ω,Q =
qp
((αj)j
):= sup
np
n∑j=1
αjxj
, for all p ∈ P
.
Theorem
If E is a complete lcs then (∧,Q) is a complete lcs.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 9 / 22
![Page 21: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/21.jpg)
Atomic Decompositions
Definition
A lcs E has the bounded approximation property (BAP) if there exists(Aj)j∈J ⊂ L (E ,E ) a net which is equicontinuous whit dim(Aj (E )) <∞for every j ∈ J and limj∈J Aj (x) = x for every x ∈ E . In other words,Aj → I in Ls (E ).
Characterization (Bessaga, Pe lczynski)
Let E be a separable Frechet space, the following are equivalent:
1 The bounded approximation property holds in E .
2 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
3 E admits an atomic decomposition.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 10 / 22
![Page 22: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/22.jpg)
Atomic Decompositions
Definition
A lcs E has the bounded approximation property (BAP) if there exists(Aj)j∈J ⊂ L (E ,E ) a net which is equicontinuous whit dim(Aj (E )) <∞for every j ∈ J and limj∈J Aj (x) = x for every x ∈ E . In other words,Aj → I in Ls (E ).
Characterization (Bessaga, Pe lczynski)
Let E be a separable Frechet space, the following are equivalent:
1 The bounded approximation property holds in E .
2 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
3 E admits an atomic decomposition.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 10 / 22
![Page 23: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/23.jpg)
Atomic Decompositions
Definition
A lcs E has the bounded approximation property (BAP) if there exists(Aj)j∈J ⊂ L (E ,E ) a net which is equicontinuous whit dim(Aj (E )) <∞for every j ∈ J and limj∈J Aj (x) = x for every x ∈ E . In other words,Aj → I in Ls (E ).
Characterization (Bessaga, Pe lczynski)
Let E be a separable Frechet space, the following are equivalent:
1 The bounded approximation property holds in E .
2 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
3 E admits an atomic decomposition.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 10 / 22
![Page 24: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/24.jpg)
Atomic Decompositions
Definition
A lcs E has the bounded approximation property (BAP) if there exists(Aj)j∈J ⊂ L (E ,E ) a net which is equicontinuous whit dim(Aj (E )) <∞for every j ∈ J and limj∈J Aj (x) = x for every x ∈ E . In other words,Aj → I in Ls (E ).
Characterization (Bessaga, Pe lczynski)
Let E be a separable Frechet space, the following are equivalent:
1 The bounded approximation property holds in E .
2 E is isomorphic to a complemented subspace of a complete sequencespace with Schauder basis.
3 E admits an atomic decomposition.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 10 / 22
![Page 25: Atomic decompositions in locally convex spacesiumpa.upv.es/wp-content/uploads/2016/05/Atomic... · Let E be a Hausdor locally convex space. De nition Let fx jg1 j=1 ˆE and let fx](https://reader034.vdocument.in/reader034/viewer/2022050608/5faf6e9d709f921ab938870d/html5/thumbnails/25.jpg)
Duality
Outline
1 Introduction
2 Atomic Decompositions
3 Duality
4 Unconditional atomic decompostions
5 Example
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 11 / 22
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Duality
Shrinking Atomic Decompositions
We define a linear operator T : E → E as Tn (x) :=∑∞
j=n+1 x ′j (x) xj .
Definition
An atomic decomposition({x ′j}, {xj}
)is shrinking if for all x ′ ∈ E ′,
limn→∞ x ′ ◦ Tn = 0 ( the limit is taken in the topologyE ′β = (E ′, β (E ′,E )).
Characterization
The following are equivalent:
1
({x ′j}, {xj}
)is a shrinking atomic decomposition.
2
({xj}, {x ′j}
)is an atomic decomposition for E ′β.
3 For all x ′ ∈ E ′,∑∞
j=1 x ′ (xj) x ′j is convergent in E ′β.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 12 / 22
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Duality
Shrinking Atomic Decompositions
We define a linear operator T : E → E as Tn (x) :=∑∞
j=n+1 x ′j (x) xj .
Definition
An atomic decomposition({x ′j}, {xj}
)is shrinking if for all x ′ ∈ E ′,
limn→∞ x ′ ◦ Tn = 0 ( the limit is taken in the topologyE ′β = (E ′, β (E ′,E )).
Characterization
The following are equivalent:
1
({x ′j}, {xj}
)is a shrinking atomic decomposition.
2
({xj}, {x ′j}
)is an atomic decomposition for E ′β.
3 For all x ′ ∈ E ′,∑∞
j=1 x ′ (xj) x ′j is convergent in E ′β.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 12 / 22
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Duality
Shrinking Atomic Decompositions
We define a linear operator T : E → E as Tn (x) :=∑∞
j=n+1 x ′j (x) xj .
Definition
An atomic decomposition({x ′j}, {xj}
)is shrinking if for all x ′ ∈ E ′,
limn→∞ x ′ ◦ Tn = 0 ( the limit is taken in the topologyE ′β = (E ′, β (E ′,E )).
Characterization
The following are equivalent:
1
({x ′j}, {xj}
)is a shrinking atomic decomposition.
2
({xj}, {x ′j}
)is an atomic decomposition for E ′β.
3 For all x ′ ∈ E ′,∑∞
j=1 x ′ (xj) x ′j is convergent in E ′β.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 12 / 22
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Duality
Shrinking Atomic Decompositions
We define a linear operator T : E → E as Tn (x) :=∑∞
j=n+1 x ′j (x) xj .
Definition
An atomic decomposition({x ′j}, {xj}
)is shrinking if for all x ′ ∈ E ′,
limn→∞ x ′ ◦ Tn = 0 ( the limit is taken in the topologyE ′β = (E ′, β (E ′,E )).
Characterization
The following are equivalent:
1
({x ′j}, {xj}
)is a shrinking atomic decomposition.
2
({xj}, {x ′j}
)is an atomic decomposition for E ′β.
3 For all x ′ ∈ E ′,∑∞
j=1 x ′ (xj) x ′j is convergent in E ′β.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 12 / 22
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Duality
Boundedly Complete Atomic Decompositions
Definition
An atomic decomposition({x ′j}, {xj}
)is boundedly complete if for all
x ′′ ∈ E ′′β , the series∑∞
j=1 x ′j (x ′′) xj converges in E .
Proposition
Let E be a barrelled lcs with a Schauder basis {xj}j ;({x ′j}, {xj}
)is a
boundedly complete atomic decomposition if and only if for every
(αj)j ⊂ K such that(∑k
j=1 αjxj
)k
is bounded, then∑∞
j=1 αjxj is
convergent.
Proposition
If({x ′j}, {xj}
)is a boundedly complete atomic decomposition for a
barrelled and complete lcs E with E ′′β barrelled, then E is complemented inits bidual E ′′β .
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 13 / 22
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Duality
Boundedly Complete Atomic Decompositions
Definition
An atomic decomposition({x ′j}, {xj}
)is boundedly complete if for all
x ′′ ∈ E ′′β , the series∑∞
j=1 x ′j (x ′′) xj converges in E .
Proposition
Let E be a barrelled lcs with a Schauder basis {xj}j ;({x ′j}, {xj}
)is a
boundedly complete atomic decomposition if and only if for every
(αj)j ⊂ K such that(∑k
j=1 αjxj
)k
is bounded, then∑∞
j=1 αjxj is
convergent.
Proposition
If({x ′j}, {xj}
)is a boundedly complete atomic decomposition for a
barrelled and complete lcs E with E ′′β barrelled, then E is complemented inits bidual E ′′β .
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 13 / 22
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Duality
Boundedly Complete Atomic Decompositions
Definition
An atomic decomposition({x ′j}, {xj}
)is boundedly complete if for all
x ′′ ∈ E ′′β , the series∑∞
j=1 x ′j (x ′′) xj converges in E .
Proposition
Let E be a barrelled lcs with a Schauder basis {xj}j ;({x ′j}, {xj}
)is a
boundedly complete atomic decomposition if and only if for every
(αj)j ⊂ K such that(∑k
j=1 αjxj
)k
is bounded, then∑∞
j=1 αjxj is
convergent.
Proposition
If({x ′j}, {xj}
)is a boundedly complete atomic decomposition for a
barrelled and complete lcs E with E ′′β barrelled, then E is complemented inits bidual E ′′β .
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 13 / 22
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Duality
Properties
Proposition
Let E be a lcs and let({x ′j}, {xj}
)be a shrinking atomic decomposition
of E . Then({xj}, {x ′j}
)is a boundedly complete atomic decomposition of
E ′.
Theorem
Let({x ′j}, {xj}
)be an atomic decomposition of a (barrelled) lcs E which
is shrinking and boundedly complete, then E is (semi-)reflexive.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 14 / 22
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Duality
Properties
Proposition
Let E be a lcs and let({x ′j}, {xj}
)be a shrinking atomic decomposition
of E . Then({xj}, {x ′j}
)is a boundedly complete atomic decomposition of
E ′.
Theorem
Let({x ′j}, {xj}
)be an atomic decomposition of a (barrelled) lcs E which
is shrinking and boundedly complete, then E is (semi-)reflexive.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 14 / 22
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Unconditional atomic decompostions
Outline
1 Introduction
2 Atomic Decompositions
3 Duality
4 Unconditional atomic decompostions
5 Example
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 15 / 22
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Unconditional atomic decompostions
Definition
Definition
An atomic decomposition({x ′j}, {xj}
)for a lcs E is said to be
unconditional if for every x ∈ E , its series expansion∑∞
j=1 x ′j (x) xjconverges unconditionally, that is, x =
∑∞j=1 x ′j (x) xj with unconditional
convergence.
McArthur, Retherford, 1969
If a series∑∞
j=1 xj converges unconditionally, then, for every boundedsequence of scalars {aj}, the series
∑∞j=1 ajxj converges and the operator
`∞ −→ E{aj} −→
∑∞j=1 ajxj ;
is a continuous linear operator.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 16 / 22
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Unconditional atomic decompostions
Definition
Definition
An atomic decomposition({x ′j}, {xj}
)for a lcs E is said to be
unconditional if for every x ∈ E , its series expansion∑∞
j=1 x ′j (x) xjconverges unconditionally, that is, x =
∑∞j=1 x ′j (x) xj with unconditional
convergence.
McArthur, Retherford, 1969
If a series∑∞
j=1 xj converges unconditionally, then, for every boundedsequence of scalars {aj}, the series
∑∞j=1 ajxj converges and the operator
`∞ −→ E{aj} −→
∑∞j=1 ajxj ;
is a continuous linear operator.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 16 / 22
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Unconditional atomic decompostions
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an unconditional atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as unconditional Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with unconditional Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 17 / 22
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Unconditional atomic decompostions
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an unconditional atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as unconditional Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with unconditional Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 17 / 22
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Unconditional atomic decompostions
Characterization
Let E be a barrelled and complete lcs, the following are equivalent:
1 E admits an unconditional atomic decomposition.
2 E is isomorphic to a complemented subspace of a complete sequencespace with the canonical unit vectors as unconditional Schauder basis.
3 E is isomorphic to a complemented subspace of a complete sequencespace with unconditional Schauder basis.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 17 / 22
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Example
Outline
1 Introduction
2 Atomic Decompositions
3 Duality
4 Unconditional atomic decompostions
5 Example
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 18 / 22
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Example
An atomic decomposition on the space C∞ (K )
Let K be a compact set in Rp, p ≥ 1, with◦K 6= ∅ such that K =
◦K .
Let C∞ (K ) := {f ∈ C∞(◦K ) : f and all its partial derivatives admit
continuous extension to K}.The Frechet space topology in C∞ (K ) is defined by the seminorms:
qn (f ) := sup{∣∣∣f (α) (x)
∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.
Remark
No system of exponencials can be a basis in C∞ ([0, 1]).
Theorem
Let K ⊂ Rp be a compact set which is the closure of its interior and let({uj}, {e ix ·λ
j})
be an unconditional atomic decomposition of C∞(K ),then there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 19 / 22
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Example
An atomic decomposition on the space C∞ (K )
Let K be a compact set in Rp, p ≥ 1, with◦K 6= ∅ such that K =
◦K .
Let C∞ (K ) := {f ∈ C∞(◦K ) : f and all its partial derivatives admit
continuous extension to K}.
The Frechet space topology in C∞ (K ) is defined by the seminorms:
qn (f ) := sup{∣∣∣f (α) (x)
∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.
Remark
No system of exponencials can be a basis in C∞ ([0, 1]).
Theorem
Let K ⊂ Rp be a compact set which is the closure of its interior and let({uj}, {e ix ·λ
j})
be an unconditional atomic decomposition of C∞(K ),then there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 19 / 22
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Example
An atomic decomposition on the space C∞ (K )
Let K be a compact set in Rp, p ≥ 1, with◦K 6= ∅ such that K =
◦K .
Let C∞ (K ) := {f ∈ C∞(◦K ) : f and all its partial derivatives admit
continuous extension to K}.The Frechet space topology in C∞ (K ) is defined by the seminorms:
qn (f ) := sup{∣∣∣f (α) (x)
∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.
Remark
No system of exponencials can be a basis in C∞ ([0, 1]).
Theorem
Let K ⊂ Rp be a compact set which is the closure of its interior and let({uj}, {e ix ·λ
j})
be an unconditional atomic decomposition of C∞(K ),then there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 19 / 22
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Example
An atomic decomposition on the space C∞ (K )
Let K be a compact set in Rp, p ≥ 1, with◦K 6= ∅ such that K =
◦K .
Let C∞ (K ) := {f ∈ C∞(◦K ) : f and all its partial derivatives admit
continuous extension to K}.The Frechet space topology in C∞ (K ) is defined by the seminorms:
qn (f ) := sup{∣∣∣f (α) (x)
∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.
Remark
No system of exponencials can be a basis in C∞ ([0, 1]).
Theorem
Let K ⊂ Rp be a compact set which is the closure of its interior and let({uj}, {e ix ·λ
j})
be an unconditional atomic decomposition of C∞(K ),then there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 19 / 22
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Example
An atomic decomposition on the space C∞ (K )
Let K be a compact set in Rp, p ≥ 1, with◦K 6= ∅ such that K =
◦K .
Let C∞ (K ) := {f ∈ C∞(◦K ) : f and all its partial derivatives admit
continuous extension to K}.The Frechet space topology in C∞ (K ) is defined by the seminorms:
qn (f ) := sup{∣∣∣f (α) (x)
∣∣∣ : x ∈ K , |α| ≤ n}, n ∈ N0.
Remark
No system of exponencials can be a basis in C∞ ([0, 1]).
Theorem
Let K ⊂ Rp be a compact set which is the closure of its interior and let({uj}, {e ix ·λ
j})
be an unconditional atomic decomposition of C∞(K ),then there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp).
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 19 / 22
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Example
An atomic decomposition on the space C∞ (K )
Theorem
Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences
(λj)⊂ Rp and
(uj) ∈ C∞ (K )′ such that({uj} ,
{e2πix ·λ
j})
is an atomic decomposition
for C∞ (K ).
Let M > 0 such that K ⊂ [−M,M]p
Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.
Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).
Then we take u (f ) = aj (Hf ).
The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 20 / 22
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Example
An atomic decomposition on the space C∞ (K )
Theorem
Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences
(λj)⊂ Rp and
(uj) ∈ C∞ (K )′ such that({uj} ,
{e2πix ·λ
j})
is an atomic decomposition
for C∞ (K ).
Let M > 0 such that K ⊂ [−M,M]p
Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.
Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).
Then we take u (f ) = aj (Hf ).
The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 20 / 22
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Example
An atomic decomposition on the space C∞ (K )
Theorem
Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences
(λj)⊂ Rp and
(uj) ∈ C∞ (K )′ such that({uj} ,
{e2πix ·λ
j})
is an atomic decomposition
for C∞ (K ).
Let M > 0 such that K ⊂ [−M,M]p
Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.
Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).
Then we take u (f ) = aj (Hf ).
The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 20 / 22
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Example
An atomic decomposition on the space C∞ (K )
Theorem
Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences
(λj)⊂ Rp and
(uj) ∈ C∞ (K )′ such that({uj} ,
{e2πix ·λ
j})
is an atomic decomposition
for C∞ (K ).
Let M > 0 such that K ⊂ [−M,M]p
Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.
Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).
Then we take u (f ) = aj (Hf ).
The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 20 / 22
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Example
An atomic decomposition on the space C∞ (K )
Theorem
Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences
(λj)⊂ Rp and
(uj) ∈ C∞ (K )′ such that({uj} ,
{e2πix ·λ
j})
is an atomic decomposition
for C∞ (K ).
Let M > 0 such that K ⊂ [−M,M]p
Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.
Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).
Then we take u (f ) = aj (Hf ).
The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 20 / 22
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Example
An atomic decomposition on the space C∞ (K )
Theorem
Let us assume that there exists a continuous linear extension operatorT : C∞ (K )→ C∞ (Rp). Then there are sequences
(λj)⊂ Rp and
(uj) ∈ C∞ (K )′ such that({uj} ,
{e2πix ·λ
j})
is an atomic decomposition
for C∞ (K ).
Let M > 0 such that K ⊂ [−M,M]p
Choosing φ ∈ D ([−2M, 2M]p) such that φ ≡ 1 on a neighborhood of[−M,M]p.
Let f ∈ C∞ (K ) we define Hf = φ (T (f )) ∈ D (]−2M, 2M[p).
Then we take u (f ) = aj (Hf ).
The atomic decomposition of C∞ (K ) is shrinking and boundedlycomplete since C∞ (K ) is a Montel space.
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 20 / 22
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Bibliography
Bibliography
1 Bonet, Jose and Perez Carreras, Pedro Barrelled locally convexspaces, North-Holland Publishing Co. (1987).
2 Carando, Daniel and Lassalle, Silvia Duality, reflexivity and atomicdecompositions in Banach spaces, Studia Math. 191 (2009.1), 67–80.
3 Casazza, Pete and Christensen, Ole and Stoeva, Diana T.Frame expansions in separable Banach spaces, J. Math. Anal. Appl.307 (2005.2), 710–723.
4 Korobeınik, Yu. F. On absolutely representing systems in spaces ofinfinitely differentiable functions, J. Math. Anal. Appl. 139 (2000.2),175–188.
5 Ribera, Juan M. Atomic Decompositions in Locally Convex Spaces,2012 (Preprint) .
Juan Miguel Ribera Puchades (UPV) Atomic decompositions in lcs Alicante, May 2012 21 / 22
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Thanks for your attention