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

## Document Type

Article

## Publication Date

2017

## Department/School

Mathematics

## Publication Title

*Mathematical Logic Quarterly*

## Abstract

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 <2ℵ0 ”, thus the latter statement is true in every Fraenkel-Mostowski model of ZFA; “No infinite-dimensional Banach space has a Hamel basis of cardinality <2ℵ0 ” is not provable in ZF; “No infinite-dimensional Banach space has a well-orderable Hamel basis of cardinality <2ℵ0 ” is provable in AC ℵ0fin (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 + αx|| 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 ball BX = {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 (<2ℵ0 ” 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 ACℵ0 (the Axiom of Countable Choice) and ACfinℵ0 DC implies “No infinite-dimensional Banach space can be written as a denumerable union of closed proper subspaces” which in turn implies ACℵ0 “Every infinite-dimensional Banach space has a denumerable linearly independent subset” is a theorem of ZF +ACℵ0, but not a theorem of ZF; and “Every infinite-dimensional Banach space has a linearly independent subset of cardinality ≥ 2ℵ0 ” implies “every Dedekind-finite set is finite”.

## Link to Published Version

## Recommended Citation

Howard, P., & Tachtsis, E. (2017). On infinite-dimensional Banach spaces and weak forms of the axiom of choice. *Mathematical Logic Quarterly, 63*(6), 509–535. https://doi.org/10.1002/malq.201600027