The equivalency of the Banach-Alaoglu theorem and the axiom of choice

Authors

  • Nand Kishor Kumar Trichandra Campus, Tribhuvan University, Nepal
  • Sher Singh Raikhola Bhaktapur Multiple Campus, Tribhuvan University, Nepal

DOI:

https://doi.org/10.3126/cognition.v6i1.64440

Keywords:

Banach-Alaoglu theorem, Tychonoff's theorem, Axiom of choice, Compactness, Dual space

Abstract

The Axiom of Choice can be used to prove the Banach-Alaoglu theorem, while a weakened form of the Axiom of Choice is required to prove the full strength of the Hahn-Banach theorem, which is equivalent to the Banach-Alaoglu theorem. Although the Banach-Alaoglu theorem and the full- strength Axiom of Choice are not exactly equivalent, they are closely related.

The Banach-Alaoglu theorem is a compactness theorem whose proof mainly depends on Tychonoff's theorem. In this article, Banach-Alaoglu theorem is equivalent to the axiom of choice, has been proved.

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Author Biographies

Nand Kishor Kumar, Trichandra Campus, Tribhuvan University, Nepal

Asst. Professor of Mathematics

 

Sher Singh Raikhola, Bhaktapur Multiple Campus, Tribhuvan University, Nepal

Asst. Professor of Mathematics

 

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Published

2024-04-08

How to Cite

Kumar, . N. K., & Raikhola, S. S. (2024). The equivalency of the Banach-Alaoglu theorem and the axiom of choice . Cognition, 6(1), 60–64. https://doi.org/10.3126/cognition.v6i1.64440

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Section

Articles