Locally convex approach spaces

Abstract

We continue the investigation of suitable structures for quantified functional analysis, by looking at the notion of local convexity in the setting of approach vector spaces as introduced in [6]. We prove that the locally convex objects are exactly the ones generated (in the usual approach sense) by collections of seminorms. Furthermore, we construct a quantified version of the projective tensor product and show that the locally convex objects admitting a decent exponential law with respect to it are precisely the seminormed spaces.

Keywords

• Approach vector space
• Topological vector space
• Locally convex space
• Locally convex approach space
• Minkowski functional
• Minkowski system
• Projective tensor product