Nonseparable spaceability and strong algebrability of sets of continuous singular functions

TL;DR

This paper demonstrates that the set of strongly singular functions within the Banach algebra of continuous functions of bounded variation is large in a spaceable sense and contains large algebraic structures, introducing a new criterion for strong c-algebrability.

Contribution

It establishes nonseparable spaceability and strong c-algebrability of sets of singular functions in CBV, along with a new criterion for strong c-algebrability.

Findings

The set of strongly singular functions in CBV is nonseparably spaceable.

Certain families of singular functions are strongly c-algebrable.

A new general criterion for strong c-algebrability is introduced.

Abstract

Let CBV denote the Banach algebra of all continuous real-valued functions of bounded variation, defined in [0,1]. We show that the set of strongly singular functions in CBV is nonseparably spaceable. We also prove that certain families of singular functions constitute strongly c-algebrable sets. The argument is based on a new general criterion of strong c-algebrability.