Locally convex inductive limit cones

TL;DR

This paper introduces a new finest order on inductive limits of ordered cones, explores their topologies, and establishes embeddings between various related structures, advancing the understanding of cone topologies.

Contribution

It defines the finest order on inductive limits of cones and investigates their topological and duality properties, providing new insights into cone theory.

Findings

Established embeddings between direct sums, inductive limits, and duals.

Analyzed weak and pointwise convergence topologies in inductive limits.

Defined the finest order making linear maps monotone.

Abstract

We define the finest order on inductive limits of ordered cones which makes the linear mappings monotone and gives rise to the definition of inductive limit topologies for cones. Using the polars of neighborhoods, we establish embeddings between direct sums, inductive limits and their duals. These lead us to investigate the weak topologies and the topologies of pointwise convergence in inductive limits.