Algebraic structure of continuous, unbounded and integrable functions

TL;DR

This paper investigates the algebraic and linear properties of unbounded continuous integrable functions on [0,+∞), analyzing their growth, smoothness, and convergence behaviors.

Contribution

It provides a detailed analysis of the size, growth, and convergence properties of unbounded integrable functions, highlighting their algebraic structure and behavior.

Findings

Large linear and algebraic size of unbounded continuous integrable functions

Analysis of growth speed and smoothness of these functions

Convergence rates to zero in various norms

Abstract

In this paper we study the large linear and algebraic size of the family of unbounded continuous and integrable functions in $[0,+\infty)$ and of the family of sequences of these functions converging to zero uniformly on compacta and in $L^1$-norm. In addition, we concentrate on the speed at which these functions grow, their smoothness and the strength of their convergence to zero.