# Simple constructions of $\mathrm{FBL}(A)$ and $\mathrm{FBL}[E]$

## TL;DR

This paper presents new constructions for free Banach lattices, showing they can be obtained as completions of free vector lattices under a maximal lattice seminorm, applicable to both sets and Banach spaces.

## Contribution

It introduces explicit completion methods for free Banach lattices derived from free vector lattices and Banach spaces, providing a unified construction approach.

## Key findings

- Construction of $ ext{FBL}(A)$ as a completion of $ ext{FVL}(A)$
- Extension of the construction to $ ext{FBL}[E]$ for Banach spaces
- Characterization of the maximal lattice seminorm $
u$

## Abstract

We show that the free Banach lattice $\mathrm{FBL}(A)$ may be constructed as the completion of $\mathrm{FVL}(A)$ with respect to the maximal lattice seminorm $\nu$ on $\mathrm{FVL}(A)$ with $\nu(a)\le 1$ for all $a\in A$. We present a similar construction for the free Banach lattice $\mathrm{FBL}[E]$ generated by a Banach space $E$.