On infinite‐dimensional Banach spaces and weak forms of the axiom of choice

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”.