Continuity of Extremal Elements in Uniformly Convex Spaces

TL;DR

This paper proves the continuous dependence of extremal elements on linear functionals in uniformly convex Banach spaces, simplifying existing proofs related to Bergman and Hardy spaces.

Contribution

It establishes the continuous dependence of extremal elements on linear functionals in uniformly convex spaces, providing a simplified proof for a known result in Bergman and Hardy spaces.

Findings

Unique extremal element exists and depends continuously on the functional

Simplified proof of extremal functional belonging to Hardy space

Clarified the relationship between extremal elements and functionals in convex spaces

Abstract

In this paper, we study the problem of finding the extremal element for a linear functional over a uniformly convex Banach space. We show that a unique extremal element exists and depends continuously on the linear functional, and vice versa. Using this, we simplify and clarify Ryabykh's proof that for any linear functional on a uniformly convex Bergman space with kernel in a certain Hardy space, the extremal functional belongs to the corresponding Hardy space.