A direct proof of F. Riesz representation Theorem

TL;DR

This paper presents a straightforward proof of the Riesz representation theorem, which characterizes linear functionals on continuous functions over compact sets, avoiding complex traditional arguments.

Contribution

It offers a direct and simplified proof of the Riesz representation theorem, enhancing understanding and accessibility of the theorem's core concepts.

Findings

Provides a direct proof of the Riesz representation theorem

Simplifies the understanding of linear functionals on C(K)

Avoids complex arguments used in traditional proofs

Abstract

A direct proof of the Riesz representation theorem is provided. This theorem characterizes the linear functionals acting on the vector space $C(K)$ of continuous functions defined on a compact subset $K$ of the real numbers $\mathbb{R}$. This proof avoids complicated arguments commonly used in generalizations of Riesz original theorem.