Notes on Spaces of Functions Converging at Infinity

TL;DR

This paper explores Banach spaces of functions that converge at infinity, including continuous, Lebesgue, and sequence spaces, and presents versions of Riesz's representation theorem for each framework.

Contribution

It provides new versions of Riesz's representation theorem tailored to various function spaces converging at infinity.

Findings

Established Riesz's representation theorem for continuous functions converging at infinity.

Extended Riesz's theorem to Lebesgue spaces with convergence at infinity.

Derived Riesz's theorem analogues for sequence spaces.

Abstract

This preprint concerns Banach spaces of functions converging at infinity. In particular, spaces of continuous functions, Lebesgue spaces and sequence spaces. In each framework we show versions of Riesz's representation theorem.