An Inequality of Uniformly Continuous Functions in Normed Spaces

TL;DR

This paper establishes a new inequality for uniformly continuous functions in normed spaces, showing that their norm can be bounded linearly by the norm of the input with specific constants.

Contribution

It introduces a novel inequality relating the norm of uniformly continuous functions to the input norm in normed spaces.

Findings

The inequality $\|f(x)ig floor ext{ is established for uniformly continuous functions.

Constants $a, b > 0$ are identified to bound the function norm.

The result provides a new tool for analyzing functions in normed spaces.

Abstract

We obtain an interesting inequalities for uniformly continuous functions in the normed spaces: $\|f(x)\|\leq a\|x\|+b$ for some $a,b> 0$.