  We show that the category of linearly topologized vector spaces over discrete
fields constitutes the correct framework for algebraic structures on Floer
homologies with field coefficients. Our case in point is the Poincar\'e duality
theorem for Rabinowitz Floer homology. We prove that Rabinowitz Floer homology
is a locally linearly compact vector space in the sense of Lefschetz, or,
equivalently, a Tate vector space in the sense of Beilinson-Feigin-Mazur.
Poincar\'e duality and the graded Frobenius algebra structure on Rabinowitz
Floer homology then hold in the topological sense. Along the way, we develop in
a largely self-contained manner the theory of linearly topologized vector
spaces, with special emphasis on duality and completed tensor products,
complementing results of Beilinson-Drinfeld, Beilinson, Rojas, Positselski, and
Esposito-Penkov.
