# The free Banach lattice generated by a Banach space

## TL;DR

This paper introduces and studies the free Banach lattice generated by a Banach space, exploring properties like the Nakano property and density character, and providing counterexamples to existing questions in the field.

## Contribution

It generalizes free Banach lattice concepts to Banach spaces and addresses open questions about lattice properties and weak compactness.

## Key findings

- Analysis of the Nakano property in free Banach lattices
- Determination of the density character of non-degenerate intervals
- Construction of a Banach lattice weakly compactly generated as a lattice but not as a Banach space

## Abstract

The free Banach lattice over a Banach space is introduced and analyzed. This generalizes the concept of free Banach lattice over a set of generators, and allows us to study the Nakano property and the density character of non-degenerate intervals on these spaces, answering some recent questions of B. de Pagter and A.W. Wickstead. Moreover, an example of a Banach lattice which is weakly compactly generated as a lattice but not as a Banach space is exhibited, thus answering a question of J. Diestel.