Maximal lineability of the set of Continuous Surjections

TL;DR

This paper proves that the set of all continuous surjective functions from  to  contains a -dimensional vector space, demonstrating a maximal lineability property in this function space.

Contribution

It establishes the maximal lineability of the set of continuous surjective functions between Euclidean spaces, showing this set contains a -dimensional vector space.

Findings

The set of continuous surjections from  to  contains a -dimensional vector space.

The result is optimal in terms of the dimension of the vector space.

Excluding the zero function, the space is maximally lineable.

Abstract

Let $m,n$ be positive integers. In this short note we prove that the set of all continuous and surjective functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$ contains (excluding the 0 function) a $\mathfrak{c}$-dimensional vector space. This result is optimal in terms of dimension.