Consider the space of signed measures over some abstract space, we know the total variation norm makes the space Banach (I guess). So are some other norms. Are there some books or literature studying such Banach spaces and their dual spaces, strong limit and weak limit?
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3$\begingroup$ Welcome to MathOverflow! I'm not sure what other norms you have in mind when you write "So are some other norms." If a norm on the space of finite signed measures over a given measure space is complete and such that the cone of all positive measures is closed, then the norm is equivalent to the total variation norm. (This is a very general phenomenon: for many spaces that carry a natural order structure there is, up to equivalence, at most one complete norm that "behaves reasonable".) $\endgroup$– Jochen GlueckCommented Jul 3, 2023 at 20:58
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$\begingroup$ there is a number of questions and answers on MO on this subject e.g. mathoverflow.net/questions/142111/… $\endgroup$– Pietro MajerCommented Jul 3, 2023 at 21:00
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3$\begingroup$ There are many books on functional analysis. As formulated, your question seems too open-ended. What is it you want to know about these kinds of Banach space? $\endgroup$– Yemon ChoiCommented Jul 3, 2023 at 21:02
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$\begingroup$ An encyclopedic reference on function spaces is Linear Operators, Part 1, General Theory by Dunford & Schwartz (Interscience, 1958). $\endgroup$– Igor KhavkineCommented Jul 3, 2023 at 23:22
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1$\begingroup$ @AndrewYing: This follows from the fact that positive linear operators between ordered Banach spaces with closed and generating cones are automatically continuous; see for instance Theorem 2.32 on page 83 of the book "Cones and Duality" (2007) by Aliprantis and Tourky. $\endgroup$– Jochen GlueckCommented Jul 4, 2023 at 12:58
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