An extension of Tychonoff’s fixed point theorem to pseudonorm adjoint topological vector spaces

Abstract

In this paper, we introduce a type of pseudonorms and quasi-pseudonorms on real vector spaces. By these concepts, we introduce the concept of pseudonorm adjoint topological vector spaces, which include locally convex topological vector spaces as special cases. By the Fan-KKM theorem, we prove a fixed point theorem in pseudonorm adjoint topological vector spaces, that is a proper extension of Tychonoff’s fixed point theorem on locally convex topological vector spaces. Then we provide an example to show that this extension is indeed a proper extension.

KEYWORDS:

• Pseudonorm
• quasi-pseudonorm
• pseudonorm adjoint topological vector space
• locally convex topological vector space
• fixed point theorem
• Fan-KKM theorem

MSC:

• 06F30
• 45G10

Acknowledgements

The author sincerely thanks the anonymous reviewers for their valuable suggestions that improved the presentation of this paper. The author is very grateful to Professors Robert Mendris, Preston Nichols, Sehie Park, Christiane Tammer, Jingfeng Tian, Yuanheng Wang, George X. Z. Yuan, and Congjun Zhang for their kind communications and comments about this paper. The author appreciates the anonymous reviewers for their valuable suggestions that improved the presentation of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).