Locally convex approach spaces

[EN] 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.Sioen, M.; Verwulgen, S. (2003). Locally convex approach spaces. Applied General Topology. 4(2):263-279. doi:10.4995/agt.2003.2031.2632794