The Canonical Lattice Isomorphism between Topologies Compatible with a Linear Space
General Topology
2023-12-01 v2 Functional Analysis
Abstract
We consider all compatible topologies of an arbitrary finite-dimensional vector space over a non-trivial valuation field whose metric completion is a locally compact space. We construct the canonical lattice isomorphism between the lattice of all compatible topologies on the vector space and the lattice of all subspaces of the vector space whose coefficient field is extended to the complete valuation field. Moreover, in this situation, we use this isomorphism to characterize the continuity of linear maps between finite-dimensional vector spaces endowed with given compatible topologies, and also, we characterize all Hausdorff compatible topologies.
Cite
@article{arxiv.1905.01880,
title = {The Canonical Lattice Isomorphism between Topologies Compatible with a Linear Space},
author = {Takanobu Aoyama},
journal= {arXiv preprint arXiv:1905.01880},
year = {2023}
}
Comments
17 pages, no figures, references and a new proposition (Proposition 5.2) are added