Separated sets in nonseparable Banach spaces Tomasz Kochanek - - PowerPoint PPT Presentation

Separated sets in nonseparable Banach spaces

Tomasz Kochanek

(based on a joint work with Tomasz Kania (Warwick)) Institute of Mathematics Polish Academy of Sciences Warsaw, Poland

Transfinite Methods in Banach Spaces and Algebras of Operators

Będlewo, July 18–22, 2016

Tomasz Kochanek (IM PAN) Separated sets 1 / 23

Basic definitions

Let X be a Banach space and A be a subset of X. We say that: A is (1+)-separated provided that x − y > 1 for all distinct x, y ∈ A; A is (1 + ε)-separated provided that x − y ≥ 1 + ε for all distinct x, y ∈ A; A is equilateral provided that the distances between any two distinct elements of A are all the same.

Tomasz Kochanek (IM PAN) Separated sets 2 / 23

Basic definitions

Let X be a Banach space and A be a subset of X. We say that: A is (1+)-separated provided that x − y > 1 for all distinct x, y ∈ A; A is (1 + ε)-separated provided that x − y ≥ 1 + ε for all distinct x, y ∈ A; A is equilateral provided that the distances between any two distinct elements of A are all the same. General question: What are the possible cardinalities of separated subsets of the unit ball of a given Banach space?

Tomasz Kochanek (IM PAN) Separated sets 2 / 23

The genesis

• F. Riesz, Über lineare Funktionalgleichungen, Acta Math. 41 (1916),

71–98.

Riesz’ lemma

If X is a normed linear space and Y is its proper subspace, then for every δ > 0 there exists a norm one vector x ∈ X with dist(x, Y ) > 1 − δ. (That is to say, one can always find an ‘almost’ orthogonal element.)

Tomasz Kochanek (IM PAN) Separated sets 3 / 23

The genesis

• F. Riesz, Über lineare Funktionalgleichungen, Acta Math. 41 (1916),

71–98.

Riesz’ lemma

If X is a normed linear space and Y is its proper subspace, then for every δ > 0 there exists a norm one vector x ∈ X with dist(x, Y ) > 1 − δ. (That is to say, one can always find an ‘almost’ orthogonal element.) This produces an infinite (1 + 0)-separated subset of the unit ball in any infinite-dimensional Banach space.

Tomasz Kochanek (IM PAN) Separated sets 3 / 23

The genesis

• F. Riesz, Über lineare Funktionalgleichungen, Acta Math. 41 (1916),

71–98.

Riesz’ lemma

If X is a normed linear space and Y is its proper subspace, then for every δ > 0 there exists a norm one vector x ∈ X with dist(x, Y ) > 1 − δ. (That is to say, one can always find an ‘almost’ orthogonal element.) This produces an infinite (1 + 0)-separated subset of the unit ball in any infinite-dimensional Banach space. k(X) = sup{|A|: A ⊆ SX, x − y > 1 for all distinct x, y ∈ A} eo(X) = sup{|A|: A ⊆ SX, x − y 1 + ε for some ε > 0 and all distinct x, y ∈ A}.

Tomasz Kochanek (IM PAN) Separated sets 3 / 23

Some classical stuff

C.A. Kottman, Subsets of the unit ball that are separated by more than

• ne, Studia Math. 53 (1975), 15–27.

Theorem(C.A. Kottman)

Every infinite-dimensional Banach space contains in its unit ball an infinite (1+)-separated set.

Tomasz Kochanek (IM PAN) Separated sets 4 / 23

Some classical stuff

C.A. Kottman, Subsets of the unit ball that are separated by more than

• ne, Studia Math. 53 (1975), 15–27.

Theorem(C.A. Kottman)

Every infinite-dimensional Banach space contains in its unit ball an infinite (1+)-separated set.

• J. Elton, E. Odell, The unit ball of every infinite-dimensional normed linear

space contains a (1 + ε)-separated sequence, Colloq. Math. 44 (1981), 105–109.

Theorem (J. Elton, E. Odell)

Every infinite-dimensional Banach space contains in its unit ball an infinite (1 + ε)-separated set for some ε > 0.

Tomasz Kochanek (IM PAN) Separated sets 4 / 23

Some classical stuff

C.A. Kottman, Subsets of the unit ball that are separated by more than

• ne, Studia Math. 53 (1975), 15–27.

Theorem(C.A. Kottman)

Every infinite-dimensional Banach space contains in its unit ball an infinite (1+)-separated set.

• J. Elton, E. Odell, The unit ball of every infinite-dimensional normed linear

space contains a (1 + ε)-separated sequence, Colloq. Math. 44 (1981), 105–109.

Theorem (J. Elton, E. Odell)

Every infinite-dimensional Banach space contains in its unit ball an infinite (1 + ε)-separated set for some ε > 0.

• A. Kryczka, S. Prus, Separated sequences in nonreflexive Banach spaces,
• Proc. Amer. Math. Soc. 129 (2000), 155-163: For nonreflexive spaces we

always have an infinite

5

√ 4-separated set.

Tomasz Kochanek (IM PAN) Separated sets 4 / 23

Equilateral sets

• P. Terenzi, Equilateral sets in Banach spaces, Boll. Un. Mat. Ital. A (7) 3

(1989), 119–124.

Theorem 3 (P. Terenzi)

There are infinite-dimensional Banach spaces which do not contain any infinite equilateral subsets.

• P. Koszmider, Uncountable equilateral sets in Banach spaces of the form

C(K), arXiv:1503.06356v2

Theorem 4 (P. Koszmider)

Under Martin’s axiom and the negation of the Continuum Hypothesis, the unit ball of every nonseparable Banach space of the form C(K) contains an uncountable 2-equilateral subset.

Tomasz Kochanek (IM PAN) Separated sets 5 / 23

Equilateral sets

• P. Terenzi, Equilateral sets in Banach spaces, Boll. Un. Mat. Ital. A (7) 3

(1989), 119–124.

Theorem 3 (P. Terenzi)

There are infinite-dimensional Banach spaces which do not contain any infinite equilateral subsets.

• P. Koszmider, Uncountable equilateral sets in Banach spaces of the form

C(K), arXiv:1503.06356v2

Theorem 4 (P. Koszmider)

Under Martin’s axiom and the negation of the Continuum Hypothesis, the unit ball of every nonseparable Banach space of the form C(K) contains an uncountable 2-equilateral subset. However, it is relatively consistent with ZFC that there exists a C(K)-space whose unit ball does not contain a (1 + ε)-separated subset for any ε > 0.

Tomasz Kochanek (IM PAN) Separated sets 5 / 23

Preliminary remarks

• 1. Note that once we have a separated set in the unit ball, we have in fact

a similar set lying on the unit sphere. This follows from the following (folklore) inequality:

• x

x − y y

• ≥ x − y

for x and y satisfying x, y ≤ 1 and x − y ≥ 1.

Tomasz Kochanek (IM PAN) Separated sets 6 / 23

Preliminary remarks

• 1. Note that once we have a separated set in the unit ball, we have in fact

a similar set lying on the unit sphere. This follows from the following (folklore) inequality:

• x

x − y y

• ≥ x − y

for x and y satisfying x, y ≤ 1 and x − y ≥ 1.

• 2. Note also that the existence of (1 + ε)-separated ‘lifts from quotients’

in the sense that the cardinality of the resulting set remains the same and instead of (1 + ε) we can have (1 + δ), for any 0 < δ < ε.

Tomasz Kochanek (IM PAN) Separated sets 6 / 23

Preliminary remarks

• 1. Note that once we have a separated set in the unit ball, we have in fact

a similar set lying on the unit sphere. This follows from the following (folklore) inequality:

• x

x − y y

• ≥ x − y

for x and y satisfying x, y ≤ 1 and x − y ≥ 1.

• 2. Note also that the existence of (1 + ε)-separated ‘lifts from quotients’

in the sense that the cardinality of the resulting set remains the same and instead of (1 + ε) we can have (1 + δ), for any 0 < δ < ε. For example, for any ε ∈ (0, 1) there are (1 + ε)-separated sets of size c in the unit ball of the Johnson–Lindenstrauss space JL as JL/c0 ∼ = ℓ2(c).

Tomasz Kochanek (IM PAN) Separated sets 6 / 23

Preliminary remarks

• 1. Note that once we have a separated set in the unit ball, we have in fact

a similar set lying on the unit sphere. This follows from the following (folklore) inequality:

• x

x − y y

• ≥ x − y

for x and y satisfying x, y ≤ 1 and x − y ≥ 1.

• 2. Note also that the existence of (1 + ε)-separated ‘lifts from quotients’

in the sense that the cardinality of the resulting set remains the same and instead of (1 + ε) we can have (1 + δ), for any 0 < δ < ε. For example, for any ε ∈ (0, 1) there are (1 + ε)-separated sets of size c in the unit ball of the Johnson–Lindenstrauss space JL as JL/c0 ∼ = ℓ2(c).

• 3. Although the unit ball of c0(ω1) obviously contains an uncountable

(1+)-separated subset, it does not contains a (1 + ε)-separated subset for any ε > 0, the fact that was already observed by Elton and Odell.

Tomasz Kochanek (IM PAN) Separated sets 6 / 23

Separated sets in nonseparable refexive-like spaces

• T. Kania, T.K., Uncountable sets of unit vectors that are separated by

more than 1, Studia Math. 232 (2016), 19–44.

Tomasz Kochanek (IM PAN) Separated sets 7 / 23

Separated sets in nonseparable refexive-like spaces

• T. Kania, T.K., Uncountable sets of unit vectors that are separated by

more than 1, Studia Math. 232 (2016), 19–44.

Theorem A (T. Kania, T.K.)

Let X be a nonseparable Banach space. (a) If X is (quasi-)reflexive, then the unit sphere of X contains an uncountable (1+)-separated subset.

Tomasz Kochanek (IM PAN) Separated sets 7 / 23

Separated sets in nonseparable refexive-like spaces

• T. Kania, T.K., Uncountable sets of unit vectors that are separated by

more than 1, Studia Math. 232 (2016), 19–44.

Theorem A (T. Kania, T.K.)

Let X be a nonseparable Banach space. (a) If X is (quasi-)reflexive, then the unit sphere of X contains an uncountable (1+)-separated subset. (b) If X is superreflexive, then the unit sphere of X contains an uncountable (1 + ε)-separated subset for some ε > 0. (This fact also extends to higher regular cardinals.)

Tomasz Kochanek (IM PAN) Separated sets 7 / 23

Separated sets in nonseparable refexive-like spaces

• T. Kania, T.K., Uncountable sets of unit vectors that are separated by

more than 1, Studia Math. 232 (2016), 19–44.

Theorem A (T. Kania, T.K.)

Let X be a nonseparable Banach space. (a) If X is (quasi-)reflexive, then the unit sphere of X contains an uncountable (1+)-separated subset. (b) If X is superreflexive, then the unit sphere of X contains an uncountable (1 + ε)-separated subset for some ε > 0. (This fact also extends to higher regular cardinals.) (c) If X is a WLD Banach space of density larger than c, then the unit sphere of X ∗ contains an uncountable (1+)-separated subset.

Tomasz Kochanek (IM PAN) Separated sets 7 / 23

Separated sets in nonseparable refexive-like spaces

• T. Kania, T.K., Uncountable sets of unit vectors that are separated by

more than 1, Studia Math. 232 (2016), 19–44.

Theorem A (T. Kania, T.K.)

Let X be a nonseparable Banach space. (a) If X is (quasi-)reflexive, then the unit sphere of X contains an uncountable (1+)-separated subset. (b) If X is superreflexive, then the unit sphere of X contains an uncountable (1 + ε)-separated subset for some ε > 0. (This fact also extends to higher regular cardinals.) (c) If X is a WLD Banach space of density larger than c, then the unit sphere of X ∗ contains an uncountable (1+)-separated subset. The first two assertions are in a sense strict, while the assumptions involved in the third one are somehow unavoidable due to the technique employed and some combinatorial reasons.

Tomasz Kochanek (IM PAN) Separated sets 7 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 1

Recall that X is called quasi-reflexive if X ∗∗/X is finite-dimensional.

Tomasz Kochanek (IM PAN) Separated sets 8 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 1

Recall that X is called quasi-reflexive if X ∗∗/X is finite-dimensional. The crucial property (Petunin, 1964 and Singer, 1963): If X is quasi-reflexive, then every total subspace M of X ∗ is necessarily norming, that is, for some c > 0 we have sup

|x∗(x)|: x∗ ∈ M ∩ BX ∗ ≥ cx

for each x ∈ X. In fact, this property characterizes quasireflexivity (Davis & Lindenstrauss, 1972).

Tomasz Kochanek (IM PAN) Separated sets 8 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 1

Recall that X is called quasi-reflexive if X ∗∗/X is finite-dimensional. The crucial property (Petunin, 1964 and Singer, 1963): If X is quasi-reflexive, then every total subspace M of X ∗ is necessarily norming, that is, for some c > 0 we have sup

|x∗(x)|: x∗ ∈ M ∩ BX ∗ ≥ cx

for each x ∈ X. In fact, this property characterizes quasireflexivity (Davis & Lindenstrauss, 1972). Folklore from linear algebra: Given finitely many linear functionals y∗, x∗

1, . . . x∗ N so that y∗ vanishes whenever all x∗ i ’s vanish, the functional y∗

must be a linear combination of x∗

i ’s.

Tomasz Kochanek (IM PAN) Separated sets 8 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 2

Let y∗ ∈ X ∗ and (x∗

n)∞ n=1 be a bounded sequence in X ∗.

Tomasz Kochanek (IM PAN) Separated sets 9 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 2

Let y∗ ∈ X ∗ and (x∗

n)∞ n=1 be a bounded sequence in X ∗. We say that y∗ is

a generalized linear combination of x∗

n’s provided that there exist:

an enumeration {z∗

1, z∗ 2, . . .} of spanQ{x∗ n : n ∈ N} ∩ BX ∗ and

a measure µ ∈ ba(PN) so that

Tomasz Kochanek (IM PAN) Separated sets 9 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 2

Let y∗ ∈ X ∗ and (x∗

n)∞ n=1 be a bounded sequence in X ∗. We say that y∗ is

a generalized linear combination of x∗

n’s provided that there exist:

an enumeration {z∗

1, z∗ 2, . . .} of spanQ{x∗ n : n ∈ N} ∩ BX ∗ and

a measure µ ∈ ba(PN) so that y∗, x =

• N

z∗

n, x µ(dn)

for each x ∈ X.

Tomasz Kochanek (IM PAN) Separated sets 9 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 2

Let y∗ ∈ X ∗ and (x∗

n)∞ n=1 be a bounded sequence in X ∗. We say that y∗ is

a generalized linear combination of x∗

n’s provided that there exist:

an enumeration {z∗

1, z∗ 2, . . .} of spanQ{x∗ n : n ∈ N} ∩ BX ∗ and

a measure µ ∈ ba(PN) so that y∗, x =

• N

z∗

n, x µ(dn)

for each x ∈ X.

• Lemma. Let X be quasireflexive, y∗ ∈ X ∗ and assume that (x∗

n)∞ n=1 is

a bounded sequence in X ∗ such that

∞

• n=1

ker(x∗

n) ⊆ ker(y∗).

Then y∗ is a generalized combination of x∗

n’s (n ∈ N).

Tomasz Kochanek (IM PAN) Separated sets 9 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 2

Let y∗ ∈ X ∗ and (x∗

n)∞ n=1 be a bounded sequence in X ∗. We say that y∗ is

a generalized linear combination of x∗

n’s provided that there exist:

an enumeration {z∗

1, z∗ 2, . . .} of spanQ{x∗ n : n ∈ N} ∩ BX ∗ and

a measure µ ∈ ba(PN) so that y∗, x =

• N

z∗

n, x µ(dn)

for each x ∈ X.

• Lemma. Let X be quasireflexive, y∗ ∈ X ∗ and assume that (x∗

n)∞ n=1 is

a bounded sequence in X ∗ such that

∞

• n=1

ker(x∗

n) ⊆ ker(y∗).

Then y∗ is a generalized combination of x∗

n’s (n ∈ N).

Important note: We can guarantee that µ ≤ 2

c y∗.

Tomasz Kochanek (IM PAN) Separated sets 9 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 3

By a result of Civin and Yood, X contains a nonseparable reflexive

• subspace. We may thus assume X is reflexive itself and hence

dens(X ∗, w∗) = dens(X) is uncountable.

Tomasz Kochanek (IM PAN) Separated sets 10 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 3

By a result of Civin and Yood, X contains a nonseparable reflexive

• subspace. We may thus assume X is reflexive itself and hence

dens(X ∗, w∗) = dens(X) is uncountable. We construct the desired sequence (xα)α<ω1 ⊂ SX together with (x∗

α)α<ω1 ⊂ SX ∗ assuming that at each step we have:

(i) x∗

β, xβ = 1 for every β < α;

(ii) xβ − xγ > 1 for all 0 ≤ β < γ < α; (iii)

γ=β ker(x∗ γ) ⊆ ker(x∗ β) for every β < α.

Tomasz Kochanek (IM PAN) Separated sets 10 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 3

By a result of Civin and Yood, X contains a nonseparable reflexive

• subspace. We may thus assume X is reflexive itself and hence

dens(X ∗, w∗) = dens(X) is uncountable. We construct the desired sequence (xα)α<ω1 ⊂ SX together with (x∗

α)α<ω1 ⊂ SX ∗ assuming that at each step we have:

(i) x∗

β, xβ = 1 for every β < α;

(ii) xβ − xγ > 1 for all 0 ≤ β < γ < α; (iii)

γ=β ker(x∗ γ) ⊆ ker(x∗ β) for every β < α.

The way of producing xα: Set N =

β<α ker(x∗ β) and pick a nonzero

vector x ∈ N (N⊥ = spanw∗{x∗

β}β<α and (X ∗, w∗) is non-separable).

Tomasz Kochanek (IM PAN) Separated sets 10 / 23

Outline of the proof of Theorem A(a)

k(X) > ω for X nonseparable, quasireflexive: Part 3

By a result of Civin and Yood, X contains a nonseparable reflexive

• subspace. We may thus assume X is reflexive itself and hence

dens(X ∗, w∗) = dens(X) is uncountable. We construct the desired sequence (xα)α<ω1 ⊂ SX together with (x∗

α)α<ω1 ⊂ SX ∗ assuming that at each step we have:

(i) x∗

β, xβ = 1 for every β < α;

(ii) xβ − xγ > 1 for all 0 ≤ β < γ < α; (iii)

γ=β ker(x∗ γ) ⊆ ker(x∗ β) for every β < α.

The way of producing xα: Set N =

β<α ker(x∗ β) and pick a nonzero

vector x ∈ N (N⊥ = spanw∗{x∗

β}β<α and (X ∗, w∗) is non-separable).

Pick K > 0 so large that y + Kx > 3

c y and define

xα = y + Kx y + Kx.

Tomasz Kochanek (IM PAN) Separated sets 10 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 1

Recall that X is called superreflexive if every ultrapower of X is reflexive.

Tomasz Kochanek (IM PAN) Separated sets 11 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 1

Recall that X is called superreflexive if every ultrapower of X is reflexive. In this case (as for any reflexive case) there is a PRI for X, say (Pα)ω≤α≤λ, where λ = dens(X).

Tomasz Kochanek (IM PAN) Separated sets 11 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 1

Recall that X is called superreflexive if every ultrapower of X is reflexive. In this case (as for any reflexive case) there is a PRI for X, say (Pα)ω≤α≤λ, where λ = dens(X). The main tool: By James’ theorem, for any ε > 0 there is some p ∈ (1, ∞) so that the operator T : X − →

• ωα<λ

(Pα+1 − Pα)(X)

• ℓp

, Tx =

Pα+1x − Pαx

• ωα<λ

has norm at most 2 + ε.

Tomasz Kochanek (IM PAN) Separated sets 11 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 1

Recall that X is called superreflexive if every ultrapower of X is reflexive. In this case (as for any reflexive case) there is a PRI for X, say (Pα)ω≤α≤λ, where λ = dens(X). The main tool: By James’ theorem, for any ε > 0 there is some p ∈ (1, ∞) so that the operator T : X − →

• ωα<λ

(Pα+1 − Pα)(X)

• ℓp

, Tx =

Pα+1x − Pαx

• ωα<λ

has norm at most 2 + ε. More precisely, for any constants 0 < c < 1/(2K), C > 1 there are exponents 1 < q < p < ∞ such that for every normalised basic sequence (en)∞

n=1 ⊂ X with basis constant K, and

any scalars (an)N

n=1, we have

c ·

N

• n=1

|an|p

• 1/p

≤

• N
• n=1

anen

• C ·

N

• n=1

|an|q

• 1/q

.

Tomasz Kochanek (IM PAN) Separated sets 11 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 2

We can assume that λ = dens(X) is a regular cardinal and has uncountable cofinality.

• Lemma. Then there exists a positive constant γmin such that for each

closed subspace Y ⊆ X with d(X/Y ) < d(X) we have T|Y γmin.

Tomasz Kochanek (IM PAN) Separated sets 12 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 2

We can assume that λ = dens(X) is a regular cardinal and has uncountable cofinality.

• Lemma. Then there exists a positive constant γmin such that for each

closed subspace Y ⊆ X with d(X/Y ) < d(X) we have T|Y γmin. Pick c < 1

2 and let p ∈ (1, ∞) as above. We shall say that the operator T

is bounded by a pair (γ, δ) if T ≤ δ and γ ≤ T|Y for every subspace Y ⊆ X with d(X/Y ) < λ.

Tomasz Kochanek (IM PAN) Separated sets 12 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 2

We can assume that λ = dens(X) is a regular cardinal and has uncountable cofinality.

• Lemma. Then there exists a positive constant γmin such that for each

closed subspace Y ⊆ X with d(X/Y ) < d(X) we have T|Y γmin. Pick c < 1

2 and let p ∈ (1, ∞) as above. We shall say that the operator T

is bounded by a pair (γ, δ) if T ≤ δ and γ ≤ T|Y for every subspace Y ⊆ X with d(X/Y ) < λ. Main claim. Our assertion follows whenever we know that there exists a pair (γ, δ) with γ/δ > 2−1/p and such that T is bounded by (γ, δ).

Tomasz Kochanek (IM PAN) Separated sets 12 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 2

We can assume that λ = dens(X) is a regular cardinal and has uncountable cofinality.

• Lemma. Then there exists a positive constant γmin such that for each

closed subspace Y ⊆ X with d(X/Y ) < d(X) we have T|Y γmin. Pick c < 1

2 and let p ∈ (1, ∞) as above. We shall say that the operator T

is bounded by a pair (γ, δ) if T ≤ δ and γ ≤ T|Y for every subspace Y ⊆ X with d(X/Y ) < λ. Main claim. Our assertion follows whenever we know that there exists a pair (γ, δ) with γ/δ > 2−1/p and such that T is bounded by (γ, δ). By induction, we build a sequence (xα)α<λ so that Txα ≥ γ and Txα’s have disjoint supports in our ℓp-sum. Then for distinct xα, xβ (α, β < λ) we have xα − xβ 1 δ Txα − Txβ = 1 δ

Txαp + Txβp1/p γ

δ · 21/p > 1.

Tomasz Kochanek (IM PAN) Separated sets 12 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 3

How can we guarantee that T is bounded by (γ, δ) with γ/δ > 2−1/p?

Tomasz Kochanek (IM PAN) Separated sets 13 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 3

How can we guarantee that T is bounded by (γ, δ) with γ/δ > 2−1/p? Set δ0 = 1

c . Since T ≤ δ0, T is bounded by a pair (γ0, δ0), where

γ0 = sup

γ : T|Y γ for every subspace Y ⊆ X with d(X/Y ) < λ .

Tomasz Kochanek (IM PAN) Separated sets 13 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 3

How can we guarantee that T is bounded by (γ, δ) with γ/δ > 2−1/p? Set δ0 = 1

c . Since T ≤ δ0, T is bounded by a pair (γ0, δ0), where

γ0 = sup

γ : T|Y γ for every subspace Y ⊆ X with d(X/Y ) < λ .

If γ0 > 2−1/pδ0, then we are done by the first part of the proof.

Tomasz Kochanek (IM PAN) Separated sets 13 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 3

How can we guarantee that T is bounded by (γ, δ) with γ/δ > 2−1/p? Set δ0 = 1

c . Since T ≤ δ0, T is bounded by a pair (γ0, δ0), where

γ0 = sup

γ : T|Y γ for every subspace Y ⊆ X with d(X/Y ) < λ .

If γ0 > 2−1/pδ0, then we are done by the first part of the proof. If not, then we replace X by some subspace X1 with dens(X/X1) < λ and T|X1 being estimated roughly by δ0.

Tomasz Kochanek (IM PAN) Separated sets 13 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 3

How can we guarantee that T is bounded by (γ, δ) with γ/δ > 2−1/p? Set δ0 = 1

c . Since T ≤ δ0, T is bounded by a pair (γ0, δ0), where

γ0 = sup

γ : T|Y γ for every subspace Y ⊆ X with d(X/Y ) < λ .

If γ0 > 2−1/pδ0, then we are done by the first part of the proof. If not, then we replace X by some subspace X1 with dens(X/X1) < λ and T|X1 being estimated roughly by δ0. The process terminates at a desired pair (γ, δ).

Tomasz Kochanek (IM PAN) Separated sets 13 / 23

Outline of the proof of Theorem A(b)

eo(X) = dens(X) for X superreflexive: Part 3

How can we guarantee that T is bounded by (γ, δ) with γ/δ > 2−1/p? Set δ0 = 1

c . Since T ≤ δ0, T is bounded by a pair (γ0, δ0), where

γ0 = sup

γ : T|Y γ for every subspace Y ⊆ X with d(X/Y ) < λ .

If γ0 > 2−1/pδ0, then we are done by the first part of the proof. If not, then we replace X by some subspace X1 with dens(X/X1) < λ and T|X1 being estimated roughly by δ0. The process terminates at a desired pair (γ, δ).

• Remark. The supremum in the definition of eo(X) need not be attained,

even in the case where X is reflexive. Indeed, let (pn)∞

n=1 be a sequence of

real numbers with p1 > 1 that increase to ∞ as n → ∞ and consider X =

n∈N

ℓpn(ωn)

• ℓ2

.

Tomasz Kochanek (IM PAN) Separated sets 13 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 1

Recall that X is weakly Lindelöf determined (WLD) if for some set Γ the dual ball (BX ∗, w∗) is homeomorphic to a subset of {x ∈ RΓ : the set {γ ∈ Γ: x(γ) = 0} is countable}, that is compact in the topology of pointwise convergence. We may assume that λ = dens(X) = c+.

Tomasz Kochanek (IM PAN) Separated sets 14 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 1

Recall that X is weakly Lindelöf determined (WLD) if for some set Γ the dual ball (BX ∗, w∗) is homeomorphic to a subset of {x ∈ RΓ : the set {γ ∈ Γ: x(γ) = 0} is countable}, that is compact in the topology of pointwise convergence. We may assume that λ = dens(X) = c+. We pick a 1-projection skeleton and, using a technique due to Kubiś, we produce from that a certain PRI in X which gives rise to an Auerbach system {( xα, x∗

α)}ω≤α<λ in X × X ∗.

(i) x∗

α =

xα = 1 for each ω ≤ α < λ; (ii) x∗

β,

xα = δαβ for all ω ≤ α, β < λ; (iii) xα ∈ X ∗∗ is weak∗-continuous for each ω ≤ α < λ,

Tomasz Kochanek (IM PAN) Separated sets 14 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 2

Define A =

x = (x∗,

xα)ω≤α<λ ∈ R[ω,λ) : x∗ ≤ 1 and x∗, xα = 0 for countably many α’s

.

Tomasz Kochanek (IM PAN) Separated sets 15 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 2

Define A =

x = (x∗,

xα)ω≤α<λ ∈ R[ω,λ) : x∗ ≤ 1 and x∗, xα = 0 for countably many α’s

.

(a1) eα ∈ A for every ω ≤ α < λ, where eα is the αth vector from the canonical basis of the vector space R[ω,λ); (a2) if x ∈ A, then −x ∈ A; (a3) supp(x) := {ω ≤ α < λ: x(α) = 0} is countable for every x ∈ A. (We write x(α) for x∗, xα, the αth coordinate of x.)

Tomasz Kochanek (IM PAN) Separated sets 15 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 2

Define A =

x = (x∗,

xα)ω≤α<λ ∈ R[ω,λ) : x∗ ≤ 1 and x∗, xα = 0 for countably many α’s

.

(a1) eα ∈ A for every ω ≤ α < λ, where eα is the αth vector from the canonical basis of the vector space R[ω,λ); (a2) if x ∈ A, then −x ∈ A; (a3) supp(x) := {ω ≤ α < λ: x(α) = 0} is countable for every x ∈ A. (We write x(α) for x∗, xα, the αth coordinate of x.) Assume, in search of a contradiction, that (a4) for every uncountable set B ⊆ A there exist x, y ∈ B with x = y such that x − y ∈ A.

Tomasz Kochanek (IM PAN) Separated sets 15 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 3

For any two sequences x, y ∈ A we shall say that y extends x if and only if there exists a third sequence z ∈ A such that the following conditions are satisfied: (e1) x(α) = y(α) = z(α) for each α sup supp(x); (e2) there is an ordinal number β with sup(supp(x)) < β < λ such that y(β) > 0 and z(β) = −1.

Tomasz Kochanek (IM PAN) Separated sets 16 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 3

For any two sequences x, y ∈ A we shall say that y extends x if and only if there exists a third sequence z ∈ A such that the following conditions are satisfied: (e1) x(α) = y(α) = z(α) for each α sup supp(x); (e2) there is an ordinal number β with sup(supp(x)) < β < λ such that y(β) > 0 and z(β) = −1.

• Claim. Every maximal chain contains a maximal element.

Tomasz Kochanek (IM PAN) Separated sets 16 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 3

For any two sequences x, y ∈ A we shall say that y extends x if and only if there exists a third sequence z ∈ A such that the following conditions are satisfied: (e1) x(α) = y(α) = z(α) for each α sup supp(x); (e2) there is an ordinal number β with sup(supp(x)) < β < λ such that y(β) > 0 and z(β) = −1.

• Claim. Every maximal chain contains a maximal element.

We build a sequence (xα)ω≤α<λ of maximal extensions of elements of our Auerbach system.

Tomasz Kochanek (IM PAN) Separated sets 16 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 3

For any two sequences x, y ∈ A we shall say that y extends x if and only if there exists a third sequence z ∈ A such that the following conditions are satisfied: (e1) x(α) = y(α) = z(α) for each α sup supp(x); (e2) there is an ordinal number β with sup(supp(x)) < β < λ such that y(β) > 0 and z(β) = −1.

• Claim. Every maximal chain contains a maximal element.

We build a sequence (xα)ω≤α<λ of maximal extensions of elements of our Auerbach system. Consider the coloring c: [[ω, λ)]2 → {0, 1} of all 2-element subsets of [ω, λ): c({α, β}) =

• if xα − xβ ∈ A,

1 if xα − xβ ∈ A.

Tomasz Kochanek (IM PAN) Separated sets 16 / 23

Outline of the proof of Theorem A(c)

k(X ∗) > ω for a WLD X with dens(X) > c: Part 3

For any two sequences x, y ∈ A we shall say that y extends x if and only if there exists a third sequence z ∈ A such that the following conditions are satisfied: (e1) x(α) = y(α) = z(α) for each α sup supp(x); (e2) there is an ordinal number β with sup(supp(x)) < β < λ such that y(β) > 0 and z(β) = −1.

• Claim. Every maximal chain contains a maximal element.

We build a sequence (xα)ω≤α<λ of maximal extensions of elements of our Auerbach system. Consider the coloring c: [[ω, λ)]2 → {0, 1} of all 2-element subsets of [ω, λ): c({α, β}) =

• if xα − xβ ∈ A,

1 if xα − xβ ∈ A. and use the Erd˝

• s–Rado theorem: c+ → (ω1)2

2.

Tomasz Kochanek (IM PAN) Separated sets 16 / 23

Separated sets in C(K)-spaces

S.K. Mercourakis, G. Vassiliadis, Equilateral sets in infinite dimensional Banach spaces, Proc. Amer. Math. Soc. 142 (2013), 205–212.

Tomasz Kochanek (IM PAN) Separated sets 17 / 23

Separated sets in C(K)-spaces

S.K. Mercourakis, G. Vassiliadis, Equilateral sets in infinite dimensional Banach spaces, Proc. Amer. Math. Soc. 142 (2013), 205–212. S.K. Mercourakis, G. Vassiliadis, Equilateral sets in Banach spaces of the form C(K), Studia Math. 231 (2015), 241–255.

Theorem B (T. Kania, T.K.)

Let K be a nonmetrizable compact Hausdorff space. Then the unit sphere

• f C(K) contains an uncountable (1+)-separated subset. Moreover,

(a) If K is not perfectly normal, then the unit sphere of C(K) contains an uncountable 2-equilateral set. (b) If K is perfectly normal, then the unit sphere of C(K) contains a (1+)-separated subset of cardinality equal to the density of C(K).

Tomasz Kochanek (IM PAN) Separated sets 17 / 23

Nonseparable reflexive space X with eo(X) = ω

Recall that c0(ω1) was an example of a nonseparable Banach space with eo(c0(ω1)) countable. Can we find a reflexive example?

Tomasz Kochanek (IM PAN) Separated sets 18 / 23

Nonseparable reflexive space X with eo(X) = ω

Recall that c0(ω1) was an example of a nonseparable Banach space with eo(c0(ω1)) countable. Can we find a reflexive example? First of all, what we need is a nonseparable Banach space lacking any nonseparable superreflexive subspaces.

• P. Hájek, Polynomials and injections of Banach spaces into superreflexive

spaces, Arch. Math. (Basel) 63 (1994), 39–44.

Tomasz Kochanek (IM PAN) Separated sets 18 / 23

Nonseparable reflexive space X with eo(X) = ω

Recall that c0(ω1) was an example of a nonseparable Banach space with eo(c0(ω1)) countable. Can we find a reflexive example? First of all, what we need is a nonseparable Banach space lacking any nonseparable superreflexive subspaces.

• P. Hájek, Polynomials and injections of Banach spaces into superreflexive

spaces, Arch. Math. (Basel) 63 (1994), 39–44. He started with a symmetric (original) Tsirelson space S(T ∗) and then defined X as the completion of c00(ω1) under the norm

• n
• i=1

aαieαi

• =
• n
• i=1

aαiei

• S

.

Tomasz Kochanek (IM PAN) Separated sets 18 / 23

Nonseparable reflexive space X with eo(X) = ω

Recall that c0(ω1) was an example of a nonseparable Banach space with eo(c0(ω1)) countable. Can we find a reflexive example? First of all, what we need is a nonseparable Banach space lacking any nonseparable superreflexive subspaces.

• P. Hájek, Polynomials and injections of Banach spaces into superreflexive

spaces, Arch. Math. (Basel) 63 (1994), 39–44. He started with a symmetric (original) Tsirelson space S(T ∗) and then defined X as the completion of c00(ω1) under the norm

• n
• i=1

aαieαi

• =
• n
• i=1

aαiei

• S

. Then, there is no one-to-one operator X → ℓp(ω1), for any 1 < p < ∞.

Tomasz Kochanek (IM PAN) Separated sets 18 / 23

Todorcevic’s walks and characteristics

• S. Todorcevic, Walks on ordinals and their characteristics, Progress in

Mathematics vol. 263, Birkhäuser Verlag AG, Basell–Boston–Berlin 2007.

Tomasz Kochanek (IM PAN) Separated sets 19 / 23

Todorcevic’s walks and characteristics

• S. Todorcevic, Walks on ordinals and their characteristics, Progress in

Mathematics vol. 263, Birkhäuser Verlag AG, Basell–Boston–Berlin 2007. By Todorcevic’s characteristics we mean any function ̺: [ω1]2 → ω satisfying the conditions: (1) ̺(α, γ) ≤ max{̺(α, β), ̺(β, γ)} for all α < β < γ < ω1; (2) ̺(α, β) ≤ max{̺(α, γ), ̺(β, γ)} (3) {α < β : ̺(α, β) ≤ n} is finite for all β < ω1 and n ∈ N.

Tomasz Kochanek (IM PAN) Separated sets 19 / 23

Todorcevic’s walks and characteristics

• S. Todorcevic, Walks on ordinals and their characteristics, Progress in

Mathematics vol. 263, Birkhäuser Verlag AG, Basell–Boston–Berlin 2007. By Todorcevic’s characteristics we mean any function ̺: [ω1]2 → ω satisfying the conditions: (1) ̺(α, γ) ≤ max{̺(α, β), ̺(β, γ)} for all α < β < γ < ω1; (2) ̺(α, β) ≤ max{̺(α, γ), ̺(β, γ)} (3) {α < β : ̺(α, β) ≤ n} is finite for all β < ω1 and n ∈ N.

• Lemma. For every pair A and B of uncountable families of pairwise

disjoint subsets of ω1 and every positive integer N, there exist uncountable subfamilies A0 ⊆ A and B0 ⊆ B such that for every pair a ∈ A0 and b ∈ B0 with a < b we have ̺(α, β) > N for all α ∈ a and β ∈ b.

Tomasz Kochanek (IM PAN) Separated sets 19 / 23

Transfinite Tsirelson-like constructions

S.A. Argyros, J. Lopez-Abad, S. Todorcevic, A class of Banach spaces with few non-strictly singular operators, J. Funct. Anal. 222 (2005), 306–384.

• Proposition. Let (xα)α<γ be a transfinite basis of X. Then X is reflexive

if and only if (xα)α<γ is shrinking and boundedly complete, that is, for each strictly increasing sequence (αn)∞

n=1 of ordinals smaller than γ, the

basis (xαn)∞

n=1 is shrinking and boundedly complete in the usual sense.

Tomasz Kochanek (IM PAN) Separated sets 20 / 23

Transfinite Tsirelson-like constructions

S.A. Argyros, J. Lopez-Abad, S. Todorcevic, A class of Banach spaces with few non-strictly singular operators, J. Funct. Anal. 222 (2005), 306–384.

• Proposition. Let (xα)α<γ be a transfinite basis of X. Then X is reflexive

if and only if (xα)α<γ is shrinking and boundedly complete, that is, for each strictly increasing sequence (αn)∞

n=1 of ordinals smaller than γ, the

basis (xαn)∞

n=1 is shrinking and boundedly complete in the usual sense.

Note that usually in Tsirelson-like constructions, bounded completeness of a given (canonical) basis follows from the fact that the space considered satisfies ‘lower f -estimates’.

Tomasz Kochanek (IM PAN) Separated sets 20 / 23

The construction of X

We can construct a nonseparable reflexive Banach space X = X[̺, ϕ, (1/mj, nj)∞

j=1]

without any uncountable (1 + ε)-separated set in the unit ball.

Tomasz Kochanek (IM PAN) Separated sets 21 / 23

The construction of X

We can construct a nonseparable reflexive Banach space X = X[̺, ϕ, (1/mj, nj)∞

j=1]

without any uncountable (1 + ε)-separated set in the unit ball. For any finite set F ⊂ ω1 we define the ̺-number of F as p̺(F) = max{̺(α, β): α, β ∈ F}.

Tomasz Kochanek (IM PAN) Separated sets 21 / 23

The construction of X

We can construct a nonseparable reflexive Banach space X = X[̺, ϕ, (1/mj, nj)∞

j=1]

without any uncountable (1 + ε)-separated set in the unit ball. For any finite set F ⊂ ω1 we define the ̺-number of F as p̺(F) = max{̺(α, β): α, β ∈ F}. Let ϕ: [0, ∞) → (0, 1] be a continuous convex functions so that ϕ(0) = 1 and limt→∞ ϕ(t) = 0. Using p̺ and ϕ we shall consider weights associated with any collection (E1 < . . . < Ek) of successive finite subsets of ω1.

Tomasz Kochanek (IM PAN) Separated sets 21 / 23

The construction of X

We can construct a nonseparable reflexive Banach space X = X[̺, ϕ, (1/mj, nj)∞

j=1]

without any uncountable (1 + ε)-separated set in the unit ball. For any finite set F ⊂ ω1 we define the ̺-number of F as p̺(F) = max{̺(α, β): α, β ∈ F}. Let ϕ: [0, ∞) → (0, 1] be a continuous convex functions so that ϕ(0) = 1 and limt→∞ ϕ(t) = 0. Using p̺ and ϕ we shall consider weights associated with any collection (E1 < . . . < Ek) of successive finite subsets of ω1. For any countable ordinal Γ let ΓΓ↑ stand for the collection of all strictly increasing functions θ: Γ → Γ with θ(α) > α for each α < Γ.

Tomasz Kochanek (IM PAN) Separated sets 21 / 23

The construction of X

For any pair (Γ, θ), where Γ < ω1 and θ ∈ ΓΓ↑, we pick two increasing sequences of natural numbers (mj(Γ, θ))∞

j=1 and (nj(Γ, θ))∞ j=1 so that

the latter dominates the former in an ‘appropriate way’ in terms of the behaviour of ̺ on [Γ]2.

Tomasz Kochanek (IM PAN) Separated sets 22 / 23

The construction of X

For any pair (Γ, θ), where Γ < ω1 and θ ∈ ΓΓ↑, we pick two increasing sequences of natural numbers (mj(Γ, θ))∞

j=1 and (nj(Γ, θ))∞ j=1 so that

the latter dominates the former in an ‘appropriate way’ in terms of the behaviour of ̺ on [Γ]2. There exists a unique norm on c00(ω1) satisfying the implicit formula x = x∞ ∨ sup

Γ<ω1

sup

θ∈ΓΓ ↑

sup

• 1

mj(Γ, θ)

nj(Γ,θ)

• i=1

ϕ(p̺(E1 ∪ . . . ∪ Ei))Eix: E1 < . . . < Enj(Γ,θ) < Γ

• .

We define X[̺, ϕ, (1/mj, nj)∞

j=1] as the completion of (c00(ω1), · ).

Tomasz Kochanek (IM PAN) Separated sets 22 / 23

Open problems

Is it (consistently) true that the unit ball of every nonseparable Banach space contains an uncountable (1+)-separated subset?

Tomasz Kochanek (IM PAN) Separated sets 23 / 23

Open problems

Is it (consistently) true that the unit ball of every nonseparable Banach space contains an uncountable (1+)-separated subset? One cannot find any counterexample in the class of reflexive spaces, neither in the class of C0(K)-spaces (K locally compact Hausdorff).

Tomasz Kochanek (IM PAN) Separated sets 23 / 23

Open problems

Is it (consistently) true that the unit ball of every nonseparable Banach space contains an uncountable (1+)-separated subset? One cannot find any counterexample in the class of reflexive spaces, neither in the class of C0(K)-spaces (K locally compact Hausdorff).

Proposition (P. Koszmider)

k(c0(c+)) ≤ c.

Tomasz Kochanek (IM PAN) Separated sets 23 / 23