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 ⊆ S X , � x − y � > 1 for all distinct x , y ∈ A } eo ( X ) sup {| A | : A ⊆ S X , � 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 one , 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 one , 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 one , 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 √ 5 always have an infinite 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 y � � � ≥ � x − 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 y � � � ≥ � x − 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 y � � � ≥ � x − 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 / c 0 ∼ = ℓ 2 ( 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 y � � � ≥ � x − 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 / c 0 ∼ = ℓ 2 ( c ) . 3. Although the unit ball of c 0 ( ω 1 ) 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
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