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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 $E$?

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I'd say the natural structure on $E$ is one of a Frechet space, given by the family of seminorms $\|f\|_{L^2(K_n)}$. Each local space generated by each seminorm is reflexive (Hilbert) so the resulting topology is reflexive. Then if you take the quotient by the closed subspace of constant functions you get again a reflexive space. Details in this nice set of lectures by Dietmar Vogt.

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Thank you for the nice reference! I got frightened in the subject when I found in Bourbaki EVT Chap.IV page 16 the remark that there are reflexive l.c. vector spaces E with closed subspaces M such that neither M nor E/M is reflexive. But Fréchet spaces are better than just any l.c. vector spaces - actually Bourbakis counterexample is a strict inductive limit of Fréchet spaces. –  Bernhard May 17 '11 at 10:09

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