What question did this study set out to answer?

The aim is to extend the Banach contraction principle to a broader class of topological vector spaces, specifically sequentially complete locally convex spaces.

July 17, 2026Ricerche di Matematica0 citationsOpen Access

Some extension of Banach contraction principle to some class of topological vector spaces

HPHubert Przybycień

Key Points

The aim is to extend the Banach contraction principle to a broader class of topological vector spaces, specifically sequentially complete locally convex spaces.
Introduced a notion of contraction in topological vector spaces.
Proved a fixed point theorem for mappings in sequentially complete and locally convex spaces.
Provided examples and remarks to illustrate the concepts.
Established that contraction mappings have fixed points in the specified class of topological vector spaces.
Demonstrated the applicability of the extension through various examples.
Highlighted potential implications for further mathematical analysis.

Abstract

The main purpose of this short note is to extend the Banach contraction principle to a relatively broad and easily describable class of topological vector spaces, namely sequentially complete locally convex topological vector spaces. To this end, we introduce a notion of contraction in topological vector spaces and prove a fixed point theorem for such mappings when the underlying space is sequentially complete and locally convex. We also provide several examples and remarks.

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