Bounded and unbounded polynomials and multilinear forms: Characterizing continuity

TL;DR

This paper characterizes the continuity of polynomials on normed spaces by their behavior on compact sets, explores the connection with connected sets, and investigates the lineability of non-continuous polynomials.

Contribution

It provides a characterization of polynomial continuity via compact set mappings and addresses the lineability of non-continuous polynomials in infinite-dimensional spaces.

Findings

A polynomial is continuous iff it maps compact sets into compact sets.

Partial answer to whether continuity is characterized by connected set transformations.

Results on the lineability of non-continuous polynomials and multilinear mappings.

Abstract

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered.