Some Applications of the Hahn-Banach Separation Theorem

TL;DR

This paper demonstrates how a specific geometric form of the Hahn-Banach separation theorem can be applied to derive various fundamental results in optimization, approximation, and numerical analysis.

Contribution

It introduces a unified approach using a single separation theorem to prove multiple classical theorems across different areas.

Findings

Derivation of Farkas type theorems

Existence results for positive coefficient quadrature

Characterizations of best approximations in Hilbert space

Abstract

We show that a single special separation theorem (namely, a consequence of the geometric form of the Hahn-Banach theorem) can be used to prove Farkas type theorems, existence theorems for numerical quadrature with positive coefficients, and detailed characterizations of best approximations from certain important cones in Hilbert space.