In an article of Sten Kaijser ("A note on dual Banach spaces") I find the assertion that $E = L^2_{\text{loc}}({\mathbb R})$ modulo constants is a reflexive space.

Question 1: which is the 'natural' locally convex top. vector space structure of this space? (maybe the proj. limit of $L^2(K_n)$-spaces modulo constants where $(K_n)_{n\ge 1}$ exhausts ${\mathbb R}$ compactly?).

Question 2: How can I see reflexivity of $E$?