Weak topologies for Linear Logic

TL;DR

This paper develops a denotational model of linear logic using locally convex separated topological vector spaces with weak topology, interpreting logical connectives through topological properties without double-orthogonality.

Contribution

It introduces a novel topological semantics for linear logic based on weak topologies, linking logical connectives to topological polarity without double-orthogonality.

Findings

Objects are locally convex separated topological vector spaces with weak topology

Negation is modeled as the dual space, and proofs as continuous linear functions

Connectives are interpreted via topological polarity

Abstract

We construct a denotational model of linear logic, whose objects are all the locally convex and separated topological vector spaces endowed with their weak topology. The negation is interpreted as the dual, linear proofs are interpreted as continuous linear functions, and non-linear proofs as sequences of monomials. We do not complete our constructions by a double-orthogonality operation. This yields an interpretation of the polarity of the connectives in terms of topology.