On Banach spaces whose norm-open sets are
weakly Fσ -sets
Witold Marciszewski and Roman Pol
University of Warsaw
Descriptive Set Theory in Paris
Dec 15–16, 2008
Marciszewski & Pol (University of Warsaw)
Paris 2008
1 / 17
We will concentrate on Banach spaces of continuous functions on
non-metrizable separable compact spaces.
Marciszewski & Pol (University of Warsaw)
Paris 2008
2 / 17
We will concentrate on Banach spaces of continuous functions on
non-metrizable separable compact spaces.
For a compact space K , C(K ) is the Banach space of real-valued
continuous functions on K (with the sup norm).
Marciszewski & Pol (University of Warsaw)
Paris 2008
2 / 17
We will concentrate on Banach spaces of continuous functions on
non-metrizable separable compact spaces.
For a compact space K , C(K ) is the Banach space of real-valued
continuous functions on K (with the sup norm).
Cp (K ) (Cw (K )) denotes C(K ) endowed with the topology of pointwise
convergence (the weak topology).
Marciszewski & Pol (University of Warsaw)
Paris 2008
2 / 17
We will concentrate on Banach spaces of continuous functions on
non-metrizable separable compact spaces.
For a compact space K , C(K ) is the Banach space of real-valued
continuous functions on K (with the sup norm).
Cp (K ) (Cw (K )) denotes C(K ) endowed with the topology of pointwise
convergence (the weak topology).
A Banach space E is (weak-norm)-perfect if each norm-open set in E
is a countable union of sets closed in the weak topology, i.e., the
identity map id : (E, weak ) → (E, norm) is of the first Borel class.
Marciszewski & Pol (University of Warsaw)
Paris 2008
2 / 17
We will concentrate on Banach spaces of continuous functions on
non-metrizable separable compact spaces.
For a compact space K , C(K ) is the Banach space of real-valued
continuous functions on K (with the sup norm).
Cp (K ) (Cw (K )) denotes C(K ) endowed with the topology of pointwise
convergence (the weak topology).
A Banach space E is (weak-norm)-perfect if each norm-open set in E
is a countable union of sets closed in the weak topology, i.e., the
identity map id : (E, weak ) → (E, norm) is of the first Borel class.
A topological space X is perfect if each open subset of X is an Fσ -set.
Marciszewski & Pol (University of Warsaw)
Paris 2008
2 / 17
We will concentrate on Banach spaces of continuous functions on
non-metrizable separable compact spaces.
For a compact space K , C(K ) is the Banach space of real-valued
continuous functions on K (with the sup norm).
Cp (K ) (Cw (K )) denotes C(K ) endowed with the topology of pointwise
convergence (the weak topology).
A Banach space E is (weak-norm)-perfect if each norm-open set in E
is a countable union of sets closed in the weak topology, i.e., the
identity map id : (E, weak ) → (E, norm) is of the first Borel class.
A topological space X is perfect if each open subset of X is an Fσ -set.
Remark
If a Banach space E is (weak-norm)-perfect, then (E, weak) is perfect,
and therefore there is a continuous linear injection T : E → `∞ .
Marciszewski & Pol (University of Warsaw)
Paris 2008
2 / 17
Example
The space `∞ ' C(βω) is not (weak-norm)-perfect (Cw (βω) and
Cp (βω) are not perfect).
Marciszewski & Pol (University of Warsaw)
Paris 2008
3 / 17
Example
The space `∞ ' C(βω) is not (weak-norm)-perfect (Cw (βω) and
Cp (βω) are not perfect).
Remark
All separable Banach spaces E are (weak-norm)-perfect.
Marciszewski & Pol (University of Warsaw)
Paris 2008
3 / 17
Example
The space `∞ ' C(βω) is not (weak-norm)-perfect (Cw (βω) and
Cp (βω) are not perfect).
Remark
All separable Banach spaces E are (weak-norm)-perfect.
Remark
If K is a scattered compact space with a (weak-norm)-perfect function
space C(K ), then K is separable.
Marciszewski & Pol (University of Warsaw)
Paris 2008
3 / 17
Example
The space `∞ ' C(βω) is not (weak-norm)-perfect (Cw (βω) and
Cp (βω) are not perfect).
Remark
All separable Banach spaces E are (weak-norm)-perfect.
Remark
If K is a scattered compact space with a (weak-norm)-perfect function
space C(K ), then K is separable.
Example (CH)
Let K be the compact scattered space of cardinality 2ℵ0 with
hereditarily Lindelöf function space Cw (K ), constructed by K. Kunen.
Then the space Cw (K ) is perfect, but C(K ) is not (weak-norm)-perfect.
Marciszewski & Pol (University of Warsaw)
Paris 2008
3 / 17
Problem
Does there exist in ZFC a compact space K such that Cw (K ) is
perfect, but C(K ) is not (weak-norm)-perfect?
Marciszewski & Pol (University of Warsaw)
Paris 2008
4 / 17
Problem
Does there exist in ZFC a compact space K such that Cw (K ) is
perfect, but C(K ) is not (weak-norm)-perfect?
An equivalent norm on a Banach space E is a Kadec norm if the weak
topology coincides with the norm topology on the unit sphere in this
norm.
Marciszewski & Pol (University of Warsaw)
Paris 2008
4 / 17
Problem
Does there exist in ZFC a compact space K such that Cw (K ) is
perfect, but C(K ) is not (weak-norm)-perfect?
An equivalent norm on a Banach space E is a Kadec norm if the weak
topology coincides with the norm topology on the unit sphere in this
norm. An equivalent norm on C(K ) is a τp -Kadec norm if the pointwise
topology coincides with the norm topology on the unit sphere.
Marciszewski & Pol (University of Warsaw)
Paris 2008
4 / 17
Problem
Does there exist in ZFC a compact space K such that Cw (K ) is
perfect, but C(K ) is not (weak-norm)-perfect?
An equivalent norm on a Banach space E is a Kadec norm if the weak
topology coincides with the norm topology on the unit sphere in this
norm. An equivalent norm on C(K ) is a τp -Kadec norm if the pointwise
topology coincides with the norm topology on the unit sphere.
Theorem
If a Banach space E has a Kadec renorming, then
(A) (G.A.Edgar, 77) The Borel structures generated by the norm and
by the weak topologies in E coincide,
Marciszewski & Pol (University of Warsaw)
Paris 2008
4 / 17
Problem
Does there exist in ZFC a compact space K such that Cw (K ) is
perfect, but C(K ) is not (weak-norm)-perfect?
An equivalent norm on a Banach space E is a Kadec norm if the weak
topology coincides with the norm topology on the unit sphere in this
norm. An equivalent norm on C(K ) is a τp -Kadec norm if the pointwise
topology coincides with the norm topology on the unit sphere.
Theorem
If a Banach space E has a Kadec renorming, then
(A) (G.A.Edgar, 77) The Borel structures generated by the norm and
by the weak topologies in E coincide, and moreover
(B) (L.Oncina, 00) norm-open sets in E are countable unions of
differences of weakly closed sets.
Marciszewski & Pol (University of Warsaw)
Paris 2008
4 / 17
Problem
Does there exist in ZFC a compact space K such that Cw (K ) is
perfect, but C(K ) is not (weak-norm)-perfect?
An equivalent norm on a Banach space E is a Kadec norm if the weak
topology coincides with the norm topology on the unit sphere in this
norm. An equivalent norm on C(K ) is a τp -Kadec norm if the pointwise
topology coincides with the norm topology on the unit sphere.
Theorem
If a Banach space E has a Kadec renorming, then
(A) (G.A.Edgar, 77) The Borel structures generated by the norm and
by the weak topologies in E coincide, and moreover
(B) (L.Oncina, 00) norm-open sets in E are countable unions of
differences of weakly closed sets.
Question (Edgar, 79, Oncina, 00)
Does (A) or (B) imply that E has a Kadec renorming?
Marciszewski & Pol (University of Warsaw)
Paris 2008
4 / 17
Theorem
It is consistent with ZFC that there exists a compact scattered space K
such that C(K ) is (weak-norm)-perfect, but has no Kadec renorming.
Marciszewski & Pol (University of Warsaw)
Paris 2008
5 / 17
Theorem
It is consistent with ZFC that there exists a compact scattered space K
such that C(K ) is (weak-norm)-perfect, but has no Kadec renorming.
Problem
Is it possible to find a negative answer to questions of Edgar and
Oncina in ZFC?
Marciszewski & Pol (University of Warsaw)
Paris 2008
5 / 17
Spaces with σ-discrete networks
Marciszewski & Pol (University of Warsaw)
Paris 2008
6 / 17
Spaces with σ-discrete networks
A network for a space X is a family A of subsets of X such that, for any
x ∈ X and its neighborhood U, there is A ∈ A with x ∈ A ⊂ U.
Marciszewski & Pol (University of Warsaw)
Paris 2008
6 / 17
Spaces with σ-discrete networks
A network for a space X is a family A of subsets of X such that, for any
x ∈ X and its neighborhood U, there is A ∈ A with x ∈ A ⊂ U.
The spaces which have σ-discrete networks are called σ-spaces.
Marciszewski & Pol (University of Warsaw)
Paris 2008
6 / 17
Spaces with σ-discrete networks
A network for a space X is a family A of subsets of X such that, for any
x ∈ X and its neighborhood U, there is A ∈ A with x ∈ A ⊂ U.
The spaces which have σ-discrete networks are called σ-spaces.
Notice that any regular σ-space is perfect.
Marciszewski & Pol (University of Warsaw)
Paris 2008
6 / 17
Spaces with σ-discrete networks
A network for a space X is a family A of subsets of X such that, for any
x ∈ X and its neighborhood U, there is A ∈ A with x ∈ A ⊂ U.
The spaces which have σ-discrete networks are called σ-spaces.
Notice that any regular σ-space is perfect.
Any space C(K ) which admits a τp -Kadec norm has the following JNR
property (Jayne, Namioka, and Rogers, Oncina):
for every ε > 0, Cp (K ) can be covered by countably many sets Mn
such that each Mn has a cover by relatively open sets with
norm-diameter ≤ ε.
Marciszewski & Pol (University of Warsaw)
Paris 2008
6 / 17
Spaces with σ-discrete networks
A network for a space X is a family A of subsets of X such that, for any
x ∈ X and its neighborhood U, there is A ∈ A with x ∈ A ⊂ U.
The spaces which have σ-discrete networks are called σ-spaces.
Notice that any regular σ-space is perfect.
Any space C(K ) which admits a τp -Kadec norm has the following JNR
property (Jayne, Namioka, and Rogers, Oncina):
for every ε > 0, Cp (K ) can be covered by countably many sets Mn
such that each Mn has a cover by relatively open sets with
norm-diameter ≤ ε.
Let JNRc be the stronger property obtained by demanding in addition
that the sets Mn in the JNR property are pointwise closed.
Marciszewski & Pol (University of Warsaw)
Paris 2008
6 / 17
Theorem
For every compact space K the following conditions are equivalent:
(i) C(K ) has the JNR property and Cp (K ) is perfect,
(ii) C(K ) has the JNRc property,
(iii) there is a σ-discrete collection in Cp (K ) which is a network for
C(K ).
Marciszewski & Pol (University of Warsaw)
Paris 2008
7 / 17
Theorem
For every compact space K the following conditions are equivalent:
(i) C(K ) has the JNR property and Cp (K ) is perfect,
(ii) C(K ) has the JNRc property,
(iii) there is a σ-discrete collection in Cp (K ) which is a network for
C(K ).
Remark
The conditions in the above theorem imply that C(K ) is
(weak-norm)-perfect and Cp (K ) is a σ-space.
Marciszewski & Pol (University of Warsaw)
Paris 2008
7 / 17
Theorem
For every compact space K the following conditions are equivalent:
(i) C(K ) has the JNR property and Cp (K ) is perfect,
(ii) C(K ) has the JNRc property,
(iii) there is a σ-discrete collection in Cp (K ) which is a network for
C(K ).
Remark
The conditions in the above theorem imply that C(K ) is
(weak-norm)-perfect and Cp (K ) is a σ-space.
If C(K ) satisfies JNRc and Cp (K ) is Lindelöf, then K is metrizable.
Marciszewski & Pol (University of Warsaw)
Paris 2008
7 / 17
Theorem
For every compact space K the following conditions are equivalent:
(i) C(K ) has the JNR property and Cp (K ) is perfect,
(ii) C(K ) has the JNRc property,
(iii) there is a σ-discrete collection in Cp (K ) which is a network for
C(K ).
Remark
The conditions in the above theorem imply that C(K ) is
(weak-norm)-perfect and Cp (K ) is a σ-space.
If C(K ) satisfies JNRc and Cp (K ) is Lindelöf, then K is metrizable.
Problem
Does there exist a compact space K such that Cp (K ) is a σ-space, but
C(K ) does not satisfy the conditions in the above theorem?
Marciszewski & Pol (University of Warsaw)
Paris 2008
7 / 17
Compact spaces KA associated with almost disjoint
families A in ω
Marciszewski & Pol (University of Warsaw)
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8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
Marciszewski & Pol (University of Warsaw)
Paris 2008
8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
KA = ω ∪ {pA : A ∈ A} ∪ {p}
Marciszewski & Pol (University of Warsaw)
Paris 2008
8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
KA = ω ∪ {pA : A ∈ A} ∪ {p}
points in ω are isolated
Marciszewski & Pol (University of Warsaw)
Paris 2008
8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
KA = ω ∪ {pA : A ∈ A} ∪ {p}
points in ω are isolated
basic neighborhoods of pA : {pA } ∪ (A \ F ) for finite F ⊂ ω
Marciszewski & Pol (University of Warsaw)
Paris 2008
8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
KA = ω ∪ {pA : A ∈ A} ∪ {p}
points in ω are isolated
basic neighborhoods of pA : {pA } ∪ (A \ F ) for finite F ⊂ ω
p is the “point at infinity” of the locally compact space ω ∪ {pA : A ∈ A}
Marciszewski & Pol (University of Warsaw)
Paris 2008
8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
KA = ω ∪ {pA : A ∈ A} ∪ {p}
points in ω are isolated
basic neighborhoods of pA : {pA } ∪ (A \ F ) for finite F ⊂ ω
p is the “point at infinity” of the locally compact space ω ∪ {pA : A ∈ A}
Such spaces were considered first by Aleksandrov and Urysohn and
we call them AU-compacta.
Marciszewski & Pol (University of Warsaw)
Paris 2008
8 / 17
Compact spaces KA associated with almost disjoint
families A in ω
A - an infinite almost disjoint family in ω (i.e., A ∈ A are infinite and
A ∩ B is finite for distinct A, B ∈ A)
KA = ω ∪ {pA : A ∈ A} ∪ {p}
points in ω are isolated
basic neighborhoods of pA : {pA } ∪ (A \ F ) for finite F ⊂ ω
p is the “point at infinity” of the locally compact space ω ∪ {pA : A ∈ A}
Such spaces were considered first by Aleksandrov and Urysohn and
we call them AU-compacta.
Remark
A space K is an AU-compactum if and only if K is separable compact
and |K (2) | = 1.
Marciszewski & Pol (University of Warsaw)
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Let K (2<ω ) and K (ω <ω ) be the AU-compacta associated with the
families of all branches of the Cantor tree 2<ω and the Baire tree ω <ω ,
respectively.
Marciszewski & Pol (University of Warsaw)
Paris 2008
9 / 17
Let K (2<ω ) and K (ω <ω ) be the AU-compacta associated with the
families of all branches of the Cantor tree 2<ω and the Baire tree ω <ω ,
respectively.
What can we say about the function spaces on K (2<ω ) and K (ω <ω )?
Marciszewski & Pol (University of Warsaw)
Paris 2008
9 / 17
Let K (2<ω ) and K (ω <ω ) be the AU-compacta associated with the
families of all branches of the Cantor tree 2<ω and the Baire tree ω <ω ,
respectively.
What can we say about the function spaces on K (2<ω ) and K (ω <ω )?
For a countable dense subset D of a separable compact space K ,
CD (K ) is the space C(K ) equipped with the topology of the pointwise
convergence on D.
Marciszewski & Pol (University of Warsaw)
Paris 2008
9 / 17
Let K (2<ω ) and K (ω <ω ) be the AU-compacta associated with the
families of all branches of the Cantor tree 2<ω and the Baire tree ω <ω ,
respectively.
What can we say about the function spaces on K (2<ω ) and K (ω <ω )?
For a countable dense subset D of a separable compact space K ,
CD (K ) is the space C(K ) equipped with the topology of the pointwise
convergence on D.
Theorem (M., 89)
Let K and L be separable compacta with homeomorphic function
spaces Cp (K ) and Cp (L). Then, for some countable dense sets D ⊂ K
and E ⊂ L, the spaces CD (K ) and CE (L) are homeomorphic.
Marciszewski & Pol (University of Warsaw)
Paris 2008
9 / 17
Let K (2<ω ) and K (ω <ω ) be the AU-compacta associated with the
families of all branches of the Cantor tree 2<ω and the Baire tree ω <ω ,
respectively.
What can we say about the function spaces on K (2<ω ) and K (ω <ω )?
For a countable dense subset D of a separable compact space K ,
CD (K ) is the space C(K ) equipped with the topology of the pointwise
convergence on D.
Theorem (M., 89)
Let K and L be separable compacta with homeomorphic function
spaces Cp (K ) and Cp (L). Then, for some countable dense sets D ⊂ K
and E ⊂ L, the spaces CD (K ) and CE (L) are homeomorphic.
Proposition
For every countable dense sets D ⊂ K (2<ω ) and E ⊂ K (ω <ω ), the
spaces CD (K (2<ω )) and CE (K (ω <ω )) are homeomorphic.
Marciszewski & Pol (University of Warsaw)
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Theorem
Let A be an infinite almost disjoint family of infinite subsets of ω. Then
the following are equivalent:
(i) C(KA ) is (weak-norm)-perfect,
(ii) Cw (KA ) is perfect,
(ii) Cp (KA ) is perfect,
(iv) A is contained in an Fσ -subset B of P(ω) consisting of infinite sets.
We identify P(ω) with the Cantor set 2ω .
Marciszewski & Pol (University of Warsaw)
Paris 2008
10 / 17
Theorem
Let A be an infinite almost disjoint family of infinite subsets of ω. Then
the following are equivalent:
(i) C(KA ) is (weak-norm)-perfect,
(ii) Cw (KA ) is perfect,
(ii) Cp (KA ) is perfect,
(iv) A is contained in an Fσ -subset B of P(ω) consisting of infinite sets.
We identify P(ω) with the Cantor set 2ω .
Corollary
The function space C(K (2<ω )) is (weak-norm)-perfect, while
C(K (ω <ω )) fails this property.
Marciszewski & Pol (University of Warsaw)
Paris 2008
10 / 17
Question (Arhangel’skii)
Let K be a compact space such that Cp (K ) is a σ-space. Is K
metrizable?
Marciszewski & Pol (University of Warsaw)
Paris 2008
11 / 17
Question (Arhangel’skii)
Let K be a compact space such that Cp (K ) is a σ-space. Is K
metrizable?
Remark
Since C(K (2<ω )) admits a τp -Kadec renorming and Cp (K (2<ω )) is
perfect, C(K (2<ω )) has the JNRc property, and therefore Cp (K (2<ω ))
is a σ-space.
Marciszewski & Pol (University of Warsaw)
Paris 2008
11 / 17
Question (Arhangel’skii)
Let K be a compact space such that Cp (K ) is a σ-space. Is K
metrizable?
Remark
Since C(K (2<ω )) admits a τp -Kadec renorming and Cp (K (2<ω )) is
perfect, C(K (2<ω )) has the JNRc property, and therefore Cp (K (2<ω ))
is a σ-space.
Remark
The space K (2<ω ) contains a copy L of the space K (ω <ω ). Since
Cp (K (ω <ω )) cannot be embedded into Cp (K (2<ω )), there is no
continuous extension operator e : Cp (L) → Cp (K (2<ω ))
(e : Cw (L) → Cw (K (2<ω ))).
Marciszewski & Pol (University of Warsaw)
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Remark
There exist maximal almost disjoint families A and B such that C(KA )
is (weak-norm)-perfect, while C(KB ) does not possess this property.
Marciszewski & Pol (University of Warsaw)
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12 / 17
Remark
There exist maximal almost disjoint families A and B such that C(KA )
is (weak-norm)-perfect, while C(KB ) does not possess this property.
Remark
There is a collection M of (maximal) almost disjoint families on ω, with
ℵ
|M| = 22 0 , such that no two distinct spaces Cw (KA )) (Cp (KA ))),
A ∈ M, are homeomorphic.
Marciszewski & Pol (University of Warsaw)
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12 / 17
Non-metrizable compacta with σ-spaces Cp (K )
Marciszewski & Pol (University of Warsaw)
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Non-metrizable compacta with σ-spaces Cp (K )
Theorem
For every separable compact linearly ordered space L, the Banach
space C(L) has the JNRc property, and therefore C(L) is
(weak-norm)-perfect and Cp (L) is a σ-space.
Marciszewski & Pol (University of Warsaw)
Paris 2008
13 / 17
Non-metrizable compacta with σ-spaces Cp (K )
Theorem
For every separable compact linearly ordered space L, the Banach
space C(L) has the JNRc property, and therefore C(L) is
(weak-norm)-perfect and Cp (L) is a σ-space.
Remark
If a compact space K is a continuous image of a compact space L and
C(L) has the JNRc property, then also C(K ) has JNRc .
Marciszewski & Pol (University of Warsaw)
Paris 2008
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Non-metrizable compacta with σ-spaces Cp (K )
Theorem
For every separable compact linearly ordered space L, the Banach
space C(L) has the JNRc property, and therefore C(L) is
(weak-norm)-perfect and Cp (L) is a σ-space.
Remark
If a compact space K is a continuous image of a compact space L and
C(L) has the JNRc property, then also C(K ) has JNRc .
Corollary
Let K be a separable monotonically normal compact space. Then
C(K ) has the JNRc property (C(K ) is (weak-norm)-perfect and Cp (K )
is a σ-space).
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Theorem
ω
ω
The space C(22 ) of functions on the Cantor cube 22 has the JNRc
property.
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Theorem
ω
ω
The space C(22 ) of functions on the Cantor cube 22 has the JNRc
property.
Recall that dyadic spaces are continuous images of Cantor cubes.
Marciszewski & Pol (University of Warsaw)
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Theorem
ω
ω
The space C(22 ) of functions on the Cantor cube 22 has the JNRc
property.
Recall that dyadic spaces are continuous images of Cantor cubes.
Corollary
For each separable dyadic compact space K , the function space C(K )
has the JNRc property, hence C(K ) is (weak-norm)-perfect and Cp (K )
is a σ-space.
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The Johnson-Lindenstrauss spaces
Marciszewski & Pol (University of Warsaw)
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The Johnson-Lindenstrauss spaces
Let K be an AU-compactum and let K 0 be the set of accumulation
points of K .
Marciszewski & Pol (University of Warsaw)
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The Johnson-Lindenstrauss spaces
Let K be an AU-compactum and let K 0 be the set of accumulation
points of K .
k · k∞ denotes the supremum norm in C(K ), and `2 (K 0 ) is the Hilbert
space of square-summable functions u : K 0 → R with the norm
1
P
2 2.
kuk2 =
|u(x)|
0
x∈K
Marciszewski & Pol (University of Warsaw)
Paris 2008
15 / 17
The Johnson-Lindenstrauss spaces
Let K be an AU-compactum and let K 0 be the set of accumulation
points of K .
k · k∞ denotes the supremum norm in C(K ), and `2 (K 0 ) is the Hilbert
space of square-summable functions u : K 0 → R with the norm
1
P
2 2.
kuk2 =
|u(x)|
0
x∈K
The Johnson-Lindenstrauss space associated with K is the space
JL(K ) = {f ∈ C(K ) : f |K 0 ∈ `2 (K 0 )} ,
equipped with the norm
kf k = max(kf k∞ , kf |K 0 k2 ),
Marciszewski & Pol (University of Warsaw)
f ∈ JL(K ) ,
Paris 2008
15 / 17
The Johnson-Lindenstrauss spaces
Let K be an AU-compactum and let K 0 be the set of accumulation
points of K .
k · k∞ denotes the supremum norm in C(K ), and `2 (K 0 ) is the Hilbert
space of square-summable functions u : K 0 → R with the norm
1
P
2 2.
kuk2 =
|u(x)|
0
x∈K
The Johnson-Lindenstrauss space associated with K is the space
JL(K ) = {f ∈ C(K ) : f |K 0 ∈ `2 (K 0 )} ,
equipped with the norm
kf k = max(kf k∞ , kf |K 0 k2 ),
f ∈ JL(K ) ,
Each Johnson-Lindenstrauss space admits a Kadec norm.
Marciszewski & Pol (University of Warsaw)
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Theorem
Let JL(K ) be the Johnson-Lindenstrauss space associated with an
AU-compactum K . Then JL(K ) is (weak-norm)-perfect if and only if
C(K ) has this property.
Marciszewski & Pol (University of Warsaw)
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Theorem
Let JL(K ) be the Johnson-Lindenstrauss space associated with an
AU-compactum K . Then JL(K ) is (weak-norm)-perfect if and only if
C(K ) has this property.
Corollary
The space JL(K (2<ω )) is (weak-norm)-perfect, while JL(K (ω <ω )) fails
this property.
Marciszewski & Pol (University of Warsaw)
Paris 2008
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Bibliography
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W. Marciszewski and R. Pol, On Banach spaces whose norm-open
sets are Fσ -sets in the weak topology, J. Math. Anal. Appl., in
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L. Oncina, The JNR property and the Borel structure of a Banach
space, Serdica Math. J. 26 (2000), no. 1, 13–32.
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Paris 2008
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