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On infinite‐dimensional Banach spaces and weak forms of the axiom of choice
Author(s) -
Howard Paul,
Tachtsis Eleftherios
Publication year - 2017
Publication title -
mathematical logic quarterly
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.473
H-Index - 28
eISSN - 1521-3870
pISSN - 0942-5616
DOI - 10.1002/malq.201600027
Subject(s) - mathematics , banach space , banach manifold , countable set , basis (linear algebra) , approximation property , cardinality (data modeling) , discrete mathematics , infinite dimensional vector function , interpolation space , pure mathematics , lp space , functional analysis , biochemistry , chemistry , geometry , computer science , gene , data mining
We study theorems from Functional Analysis with regard to their relationship with various weak choice principles and prove several results about them: “Every infinite‐dimensional Banach space has a well‐orderable Hamel basis” is equivalent to AC ; “ R can be well‐ordered” implies “no infinite‐dimensional Banach space has a Hamel basis of cardinality < 2 ℵ 0”, thus the latter statement is true in every Fraenkel‐Mostowski model of ZFA ; “No infinite‐dimensional Banach space has a Hamel basis of cardinality < 2 ℵ 0” is not provable in ZF ; “No infinite‐dimensional Banach space has a well‐orderable Hamel basis of cardinality < 2 ℵ 0” is provable in ZF ; AC fin ℵ 0(the Axiom of Choice for denumerable families of non‐empty finite sets) is equivalent to “no infinite‐dimensional Banach space has a Hamel basis which can be written as a denumerable union of finite sets”; Mazur's Lemma (“If X is an infinite‐dimensional Banach space, Y is a finite‐dimensional vector subspace of X , and ε > 0 , then there is a unit vector x ∈ X such that | | y | | ≤ ( 1 + ε ) | | y + α x | | for all y ∈ Y and all scalars α”) is provable in ZF ; “A real normed vector space X is finite‐dimensional if and only if its closed unit ballB X = { x ∈ X : | | x | | ≤ 1 }is compact” is provable in ZF ; DC (Principle of Dependent Choices) + “ R can be well‐ordered” does not imply the Hahn‐Banach Theorem ( HB ) in ZF ; HB and “no infinite‐dimensional Banach space has a Hamel basis of cardinality < 2 ℵ 0” are independent from each other in ZF ; “No infinite‐dimensional Banach space can be written as a denumerable union of finite‐dimensional subspaces” lies in strength between AC ℵ 0(the Axiom of Countable Choice) and AC fin ℵ 0 ; DC implies “No infinite‐dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies AC ℵ 0 ; “Every infinite‐dimensional Banach space has a denumerable linearly independent subset” is a theorem of ZF + AC ℵ 0, but not a theorem of ZF ; and “Every infinite‐dimensional Banach space has a linearly independent subset of cardinality ≥ 2 ℵ 0” implies “every Dedekind‐finite set is finite”.