# A stand-alone analysis of quasidensity

## TL;DR

This paper explores the concept of quasidensity in Banach spaces, establishing new connections with type (NI) sets, and provides sum theorems for coincidence sets, advancing the understanding of monotone multifunctions.

## Contribution

It introduces new results on quasidensity, clarifies the relationship with type (NI), and proves sum theorems for coincidence sets, independent of previous Banach SN space work.

## Key findings

- Quasidensity and type (NI) are not equivalent without monotonicity.
- New sum theorems for coincidence sets are established.
- Enhanced understanding of Fitzpatrick extension of quasidense monotone multifunctions.

## Abstract

In this paper we consider the "quasidensity" of a subset of the product of a Banach space and its dual, and give a connection between quasidense sets and sets of "type (NI)". We discuss "coincidence sets" of certain convex functions and prove two sum theorems for coincidence sets. We obtain new results on the Fitzpatrick extension of a closed quasidense monotone multifunction. The analysis in this paper is self-contained, and independent of previous work on "Banach SN spaces". This version differs from the previous version because it is shown that the (well known) equivalence of quasidensity and "type (NI)" for maximally monotone sets is not true without the monotonicity assumption and that the appendix has been moved to the end of Section 10, where it rightfully belongs.