The convex hull of a Banach-Saks set
Pedro Tradacete
Universidad Carlos III de Madrid
Joint work with J. Lopez-Abad (CSIC) and C. Ruiz-Bermejo (UCM)
Congreso de la RSME
22 de enero de 2013
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
1 / 12
Convergent sequences
Let (X , k · k) be a Banach space. (xn )n ⊂ X
xn −→ x
Cesaro
xn −→ x
⇔
kxn − xk → 0
⇔
n
1 X
xj − x → 0
n
j=1
w
xn −→ x
xn −→ x
P. Tradacete (UC3M)
⇔
⇒
x ∗ (xn − x) → 0, ∀x ∗ ∈ X ∗
Cesaro
xn −→ x
⇒
The convex hull of a Banach-Saks set
w
xn −→ x
RSME 2013
2 / 12
Convergent sequences
Let (X , k · k) be a Banach space. (xn )n ⊂ X
xn −→ x
Cesaro
xn −→ x
⇔
kxn − xk → 0
⇔
n
1 X
xj − x → 0
n
j=1
w
xn −→ x
xn −→ x
P. Tradacete (UC3M)
⇔
⇒
x ∗ (xn − x) → 0, ∀x ∗ ∈ X ∗
Cesaro
xn −→ x
⇒
The convex hull of a Banach-Saks set
w
xn −→ x
RSME 2013
2 / 12
Convergent sequences
Let (X , k · k) be a Banach space. (xn )n ⊂ X
xn −→ x
Cesaro
xn −→ x
⇔
kxn − xk → 0
⇔
n
1 X
xj − x → 0
n
j=1
w
xn −→ x
xn −→ x
P. Tradacete (UC3M)
⇔
⇒
x ∗ (xn − x) → 0, ∀x ∗ ∈ X ∗
Cesaro
xn −→ x
⇒
The convex hull of a Banach-Saks set
w
xn −→ x
RSME 2013
2 / 12
Convergent sequences
Let (X , k · k) be a Banach space. (xn )n ⊂ X
xn −→ x
Cesaro
xn −→ x
⇔
kxn − xk → 0
⇔
n
1 X
xj − x → 0
n
j=1
w
xn −→ x
xn −→ x
P. Tradacete (UC3M)
⇔
⇒
x ∗ (xn − x) → 0, ∀x ∗ ∈ X ∗
Cesaro
xn −→ x
⇒
The convex hull of a Banach-Saks set
w
xn −→ x
RSME 2013
2 / 12
Definition
A subset A ⊆ X of a Banach space is called Banach-Saks if every
sequence in A has a Cesàro convergent subsequence.
Examples:
1
The unit basis of c0 , `p , p > 1 are Banach-Saks (and weakly-null)
2
The unit basis of `1 is not (and it is not weakly-null)
3
The unit basis of the Schreier space XS is not, but it is weakly-null.
Recall XS is the completion of c00 under the norm given by
X
k(an )kXS = sup
|an |,
E∈S n∈E
where S is the class of finite sets of the form {n1 < n2 < · · · < nk } with
k ≤ n1 .
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
3 / 12
Definition
A subset A ⊆ X of a Banach space is called Banach-Saks if every
sequence in A has a Cesàro convergent subsequence.
Examples:
1
The unit basis of c0 , `p , p > 1 are Banach-Saks (and weakly-null)
2
The unit basis of `1 is not (and it is not weakly-null)
3
The unit basis of the Schreier space XS is not, but it is weakly-null.
Recall XS is the completion of c00 under the norm given by
X
k(an )kXS = sup
|an |,
E∈S n∈E
where S is the class of finite sets of the form {n1 < n2 < · · · < nk } with
k ≤ n1 .
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
3 / 12
Definition
A subset A ⊆ X of a Banach space is called Banach-Saks if every
sequence in A has a Cesàro convergent subsequence.
Examples:
1
The unit basis of c0 , `p , p > 1 are Banach-Saks (and weakly-null)
2
The unit basis of `1 is not (and it is not weakly-null)
3
The unit basis of the Schreier space XS is not, but it is weakly-null.
Recall XS is the completion of c00 under the norm given by
X
k(an )kXS = sup
|an |,
E∈S n∈E
where S is the class of finite sets of the form {n1 < n2 < · · · < nk } with
k ≤ n1 .
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
3 / 12
Definition
A subset A ⊆ X of a Banach space is called Banach-Saks if every
sequence in A has a Cesàro convergent subsequence.
Examples:
1
The unit basis of c0 , `p , p > 1 are Banach-Saks (and weakly-null)
2
The unit basis of `1 is not (and it is not weakly-null)
3
The unit basis of the Schreier space XS is not, but it is weakly-null.
Recall XS is the completion of c00 under the norm given by
X
k(an )kXS = sup
|an |,
E∈S n∈E
where S is the class of finite sets of the form {n1 < n2 < · · · < nk } with
k ≤ n1 .
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
3 / 12
Definition
A subset A ⊆ X of a Banach space is called Banach-Saks if every
sequence in A has a Cesàro convergent subsequence.
Examples:
1
The unit basis of c0 , `p , p > 1 are Banach-Saks (and weakly-null)
2
The unit basis of `1 is not (and it is not weakly-null)
3
The unit basis of the Schreier space XS is not, but it is weakly-null.
Recall XS is the completion of c00 under the norm given by
X
k(an )kXS = sup
|an |,
E∈S n∈E
where S is the class of finite sets of the form {n1 < n2 < · · · < nk } with
k ≤ n1 .
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
3 / 12
Convex hulls
A compact ⇒ A Banach-Saks ⇒ A weakly-compact.
Given A ⊂ X ,
co(A) := closure
nX
λi xi : λi ≥ 0,
i
X
o
λi ≤ 1, xi ∈ A
i
A compact ⇒ co(A) compact (Mazur).
A weakly-compact ⇒ co(A) weakly-compact (Krein-Smulian)
Question: A Banach-Saks ⇒ co(A) Banach-Saks?
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
4 / 12
Convex hulls
A compact ⇒ A Banach-Saks ⇒ A weakly-compact.
Given A ⊂ X ,
co(A) := closure
nX
λi xi : λi ≥ 0,
i
X
o
λi ≤ 1, xi ∈ A
i
A compact ⇒ co(A) compact (Mazur).
A weakly-compact ⇒ co(A) weakly-compact (Krein-Smulian)
Question: A Banach-Saks ⇒ co(A) Banach-Saks?
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
4 / 12
Convex hulls
A compact ⇒ A Banach-Saks ⇒ A weakly-compact.
Given A ⊂ X ,
co(A) := closure
nX
λi xi : λi ≥ 0,
i
X
o
λi ≤ 1, xi ∈ A
i
A compact ⇒ co(A) compact (Mazur).
A weakly-compact ⇒ co(A) weakly-compact (Krein-Smulian)
Question: A Banach-Saks ⇒ co(A) Banach-Saks?
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
4 / 12
Convex hulls
A compact ⇒ A Banach-Saks ⇒ A weakly-compact.
Given A ⊂ X ,
co(A) := closure
nX
λi xi : λi ≥ 0,
i
X
o
λi ≤ 1, xi ∈ A
i
A compact ⇒ co(A) compact (Mazur).
A weakly-compact ⇒ co(A) weakly-compact (Krein-Smulian)
Question: A Banach-Saks ⇒ co(A) Banach-Saks?
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
4 / 12
Convex hulls
A compact ⇒ A Banach-Saks ⇒ A weakly-compact.
Given A ⊂ X ,
co(A) := closure
nX
λi xi : λi ≥ 0,
i
X
o
λi ≤ 1, xi ∈ A
i
A compact ⇒ co(A) compact (Mazur).
A weakly-compact ⇒ co(A) weakly-compact (Krein-Smulian)
Question: A Banach-Saks ⇒ co(A) Banach-Saks?
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
4 / 12
Positive results
A Banach space has the weak Banach-Saks property if every weakly
convergent sequence has a Cesàro convergent subsequence.
Examples: Lp (1 ≤ p < ∞), c0 , . . .
Proposition
Let X have the weak Banach-Saks property. A ⊂ X is Banach-Saks if
and only if co(A) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
5 / 12
Positive results
A Banach space has the weak Banach-Saks property if every weakly
convergent sequence has a Cesàro convergent subsequence.
Examples: Lp (1 ≤ p < ∞), c0 , . . .
Proposition
Let X have the weak Banach-Saks property. A ⊂ X is Banach-Saks if
and only if co(A) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
5 / 12
Positive results
A sequence (xn )n in a Banach space X is weakly uniformly convergent
to x ∈ X if for every ε > 0, there is n(ε) ∈ N such that for every x ∗ ∈ X ∗
#{n ∈ N : |x ∗ (xn − x)| ≥ ε} ≤ n(ε).
Theorem (Mercourakis)
Cesaro
(xn )n converges uniformly weakly to x ⇔ ∀(xnk )k , xnk −→ x.
Proposition
If (xn )n is uniformly weakly convergent ⇒ co({xn }) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
6 / 12
Positive results
A sequence (xn )n in a Banach space X is weakly uniformly convergent
to x ∈ X if for every ε > 0, there is n(ε) ∈ N such that for every x ∗ ∈ X ∗
#{n ∈ N : |x ∗ (xn − x)| ≥ ε} ≤ n(ε).
Theorem (Mercourakis)
Cesaro
(xn )n converges uniformly weakly to x ⇔ ∀(xnk )k , xnk −→ x.
Proposition
If (xn )n is uniformly weakly convergent ⇒ co({xn }) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
6 / 12
Schreier spaces
Theorem (González-Rodrı́guez)
A ⊆ XS is Banach-Saks if and only if co(A) is Banach-Saks.
The Schreier family S can be extended by induction
S2 = S⊗S = s1 ∪· · ·∪sn : si ∈ S, s1 < · · · < sn , {min(s1 ), . . . , min(sn )} ∈ S
S3 = S2 ⊗ S
...
Sα can be defined for any countable ordinal α.
Theorem
A ⊆ XSα is Banach-Saks if and only if co(A) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
7 / 12
Schreier spaces
Theorem (González-Rodrı́guez)
A ⊆ XS is Banach-Saks if and only if co(A) is Banach-Saks.
The Schreier family S can be extended by induction
S2 = S⊗S = s1 ∪· · ·∪sn : si ∈ S, s1 < · · · < sn , {min(s1 ), . . . , min(sn )} ∈ S
S3 = S2 ⊗ S
...
Sα can be defined for any countable ordinal α.
Theorem
A ⊆ XSα is Banach-Saks if and only if co(A) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
7 / 12
Schreier spaces
Theorem (González-Rodrı́guez)
A ⊆ XS is Banach-Saks if and only if co(A) is Banach-Saks.
The Schreier family S can be extended by induction
S2 = S⊗S = s1 ∪· · ·∪sn : si ∈ S, s1 < · · · < sn , {min(s1 ), . . . , min(sn )} ∈ S
S3 = S2 ⊗ S
...
Sα can be defined for any countable ordinal α.
Theorem
A ⊆ XSα is Banach-Saks if and only if co(A) is Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
7 / 12
Towards a counterexample
Let F ⊆ [N]<ω be a compact family.
k(an )kXF = sup
X
|an |
E∈F n∈E
Definition
A family F is large in M when for every n ∈ N and N ⊆ M there is
s ∈ F such that #(s ∩ N) ≥ n.
Definition
A T -family is a compact and hereditary family F on N such that:
(1) F is never large in any M ⊆ N.
S
(2) There is a partition n In = N in finite sets In and δ > 0 such that
Gδ (F) := {t ⊆ N : there is s ∈ F with #(s ∩ In ) ≥ δ#In for all n ∈ t}
is large.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
8 / 12
Towards a counterexample
Let F ⊆ [N]<ω be a compact family.
k(an )kXF = sup
X
|an |
E∈F n∈E
Definition
A family F is large in M when for every n ∈ N and N ⊆ M there is
s ∈ F such that #(s ∩ N) ≥ n.
Definition
A T -family is a compact and hereditary family F on N such that:
(1) F is never large in any M ⊆ N.
S
(2) There is a partition n In = N in finite sets In and δ > 0 such that
Gδ (F) := {t ⊆ N : there is s ∈ F with #(s ∩ In ) ≥ δ#In for all n ∈ t}
is large.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
8 / 12
Towards a counterexample
Let F ⊆ [N]<ω be a compact family.
k(an )kXF = sup
X
|an |
E∈F n∈E
Definition
A family F is large in M when for every n ∈ N and N ⊆ M there is
s ∈ F such that #(s ∩ N) ≥ n.
Definition
A T -family is a compact and hereditary family F on N such that:
(1) F is never large in any M ⊆ N.
S
(2) There is a partition n In = N in finite sets In and δ > 0 such that
Gδ (F) := {t ⊆ N : there is s ∈ F with #(s ∩ In ) ≥ δ#In for all n ∈ t}
is large.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
8 / 12
Towards a counterexample
Let F ⊆ [N]<ω be a compact family.
k(an )kXF = sup
X
|an |
E∈F n∈E
Definition
A family F is large in M when for every n ∈ N and N ⊆ M there is
s ∈ F such that #(s ∩ N) ≥ n.
Definition
A T -family is a compact and hereditary family F on N such that:
(1) F is never large in any M ⊆ N.
S
(2) There is a partition n In = N in finite sets In and δ > 0 such that
Gδ (F) := {t ⊆ N : there is s ∈ F with #(s ∩ In ) ≥ δ#In for all n ∈ t}
is large.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
8 / 12
Towards a counterexample
Let F ⊆ [N]<ω be a compact family.
k(an )kXF = sup
X
|an |
E∈F n∈E
Definition
A family F is large in M when for every n ∈ N and N ⊆ M there is
s ∈ F such that #(s ∩ N) ≥ n.
Definition
A T -family is a compact and hereditary family F on N such that:
(1) F is never large in any M ⊆ N.
S
(2) There is a partition n In = N in finite sets In and δ > 0 such that
Gδ (F) := {t ⊆ N : there is s ∈ F with #(s ∩ In ) ≥ δ#In for all n ∈ t}
is large.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
8 / 12
Theorem
There is a T -family. In fact, for every ε > 0 there is a family F such that
1
F is not 4-large in any M.
2
G1−ε (F) = S.
Therefore, in XF every subsequence of the unit basis (un )n has a
subsequence 4-equivalent to the unit basis of c0 . In particular,
(un )n is Banach-Saks.
P
While, the sequence xn = #I1 n j∈In uj ∈ co({un }) is equivalent to the
unit basis of Schreier space XS . Thus,
(xn )n is not Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
9 / 12
Theorem
There is a T -family. In fact, for every ε > 0 there is a family F such that
1
F is not 4-large in any M.
2
G1−ε (F) = S.
Therefore, in XF every subsequence of the unit basis (un )n has a
subsequence 4-equivalent to the unit basis of c0 . In particular,
(un )n is Banach-Saks.
P
While, the sequence xn = #I1 n j∈In uj ∈ co({un }) is equivalent to the
unit basis of Schreier space XS . Thus,
(xn )n is not Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
9 / 12
Theorem
There is a T -family. In fact, for every ε > 0 there is a family F such that
1
F is not 4-large in any M.
2
G1−ε (F) = S.
Therefore, in XF every subsequence of the unit basis (un )n has a
subsequence 4-equivalent to the unit basis of c0 . In particular,
(un )n is Banach-Saks.
P
While, the sequence xn = #I1 n j∈In uj ∈ co({un }) is equivalent to the
unit basis of Schreier space XS . Thus,
(xn )n is not Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
9 / 12
Theorem
There is a T -family. In fact, for every ε > 0 there is a family F such that
1
F is not 4-large in any M.
2
G1−ε (F) = S.
Therefore, in XF every subsequence of the unit basis (un )n has a
subsequence 4-equivalent to the unit basis of c0 . In particular,
(un )n is Banach-Saks.
P
While, the sequence xn = #I1 n j∈In uj ∈ co({un }) is equivalent to the
unit basis of Schreier space XS . Thus,
(xn )n is not Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
9 / 12
Theorem
There is a T -family. In fact, for every ε > 0 there is a family F such that
1
F is not 4-large in any M.
2
G1−ε (F) = S.
Therefore, in XF every subsequence of the unit basis (un )n has a
subsequence 4-equivalent to the unit basis of c0 . In particular,
(un )n is Banach-Saks.
P
While, the sequence xn = #I1 n j∈In uj ∈ co({un }) is equivalent to the
unit basis of Schreier space XS . Thus,
(xn )n is not Banach-Saks.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
9 / 12
Proof of the Theorem:
Please, go to http://arxiv.org/abs/1209.4851
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
10 / 12
Proof of the Theorem:
Please, go to http://arxiv.org/abs/1209.4851
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
10 / 12
Proposition (Gillis)
For every ε > 0, δ > 0 and m ∈ N, there is n := n(ε, δ, m) such that for
every probability space (Ω, F, µ) and every sequence (Ai )i<n with
µ(Ai ) ≥ ε for all i < n, there is s ⊂ {1, . . . , n} with ]s = m such that
\
µ( Ai ) ≥ (1 − δ)εm .
i∈s
A key idea in the proof of our theorem is the following construction by
Erdős and Hajnal:
Let r , n ∈ N. Given i < j < n, let
Ai,j := {(ak )k<n ∈ r n : ai 6= aj }.
Clearly #Ai,j = r n−1 (r − 1). Now if s ⊆ n has cardinality ≥ r + 1, then
\
Ai,j = ∅.
{i,j}∈[s]2
This provides a counterexample for double-indexed sequences of the
expected generalization of Gillis’ result.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
11 / 12
Proposition (Gillis)
For every ε > 0, δ > 0 and m ∈ N, there is n := n(ε, δ, m) such that for
every probability space (Ω, F, µ) and every sequence (Ai )i<n with
µ(Ai ) ≥ ε for all i < n, there is s ⊂ {1, . . . , n} with ]s = m such that
\
µ( Ai ) ≥ (1 − δ)εm .
i∈s
A key idea in the proof of our theorem is the following construction by
Erdős and Hajnal:
Let r , n ∈ N. Given i < j < n, let
Ai,j := {(ak )k<n ∈ r n : ai 6= aj }.
Clearly #Ai,j = r n−1 (r − 1). Now if s ⊆ n has cardinality ≥ r + 1, then
\
Ai,j = ∅.
{i,j}∈[s]2
This provides a counterexample for double-indexed sequences of the
expected generalization of Gillis’ result.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
11 / 12
Proposition (Gillis)
For every ε > 0, δ > 0 and m ∈ N, there is n := n(ε, δ, m) such that for
every probability space (Ω, F, µ) and every sequence (Ai )i<n with
µ(Ai ) ≥ ε for all i < n, there is s ⊂ {1, . . . , n} with ]s = m such that
\
µ( Ai ) ≥ (1 − δ)εm .
i∈s
A key idea in the proof of our theorem is the following construction by
Erdős and Hajnal:
Let r , n ∈ N. Given i < j < n, let
Ai,j := {(ak )k<n ∈ r n : ai 6= aj }.
Clearly #Ai,j = r n−1 (r − 1). Now if s ⊆ n has cardinality ≥ r + 1, then
\
Ai,j = ∅.
{i,j}∈[s]2
This provides a counterexample for double-indexed sequences of the
expected generalization of Gillis’ result.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
11 / 12
Proposition (Gillis)
For every ε > 0, δ > 0 and m ∈ N, there is n := n(ε, δ, m) such that for
every probability space (Ω, F, µ) and every sequence (Ai )i<n with
µ(Ai ) ≥ ε for all i < n, there is s ⊂ {1, . . . , n} with ]s = m such that
\
µ( Ai ) ≥ (1 − δ)εm .
i∈s
A key idea in the proof of our theorem is the following construction by
Erdős and Hajnal:
Let r , n ∈ N. Given i < j < n, let
Ai,j := {(ak )k<n ∈ r n : ai 6= aj }.
Clearly #Ai,j = r n−1 (r − 1). Now if s ⊆ n has cardinality ≥ r + 1, then
\
Ai,j = ∅.
{i,j}∈[s]2
This provides a counterexample for double-indexed sequences of the
expected generalization of Gillis’ result.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
11 / 12
Proposition (Gillis)
For every ε > 0, δ > 0 and m ∈ N, there is n := n(ε, δ, m) such that for
every probability space (Ω, F, µ) and every sequence (Ai )i<n with
µ(Ai ) ≥ ε for all i < n, there is s ⊂ {1, . . . , n} with ]s = m such that
\
µ( Ai ) ≥ (1 − δ)εm .
i∈s
A key idea in the proof of our theorem is the following construction by
Erdős and Hajnal:
Let r , n ∈ N. Given i < j < n, let
Ai,j := {(ak )k<n ∈ r n : ai 6= aj }.
Clearly #Ai,j = r n−1 (r − 1). Now if s ⊆ n has cardinality ≥ r + 1, then
\
Ai,j = ∅.
{i,j}∈[s]2
This provides a counterexample for double-indexed sequences of the
expected generalization of Gillis’ result.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
11 / 12
Thank you very much for your attention.
P. Tradacete (UC3M)
The convex hull of a Banach-Saks set
RSME 2013
12 / 12