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I would like to ask for some request on the relationship between Mackey topology and the weak topology. Recently, I have read quite some statements where a property in weak topology (such as compactness) implies that in Mackey topology. The basic connection I have now is that the topology is identitical if we are talking about an equicontinuous subset. However, I have a feeling that there must be something more. The reference I got is schaefer, Topological Vector Space and Bourbaki, Topological Vector Space. However, both of the books are quite brief on this matter.

Thanks in advance.

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The basic reference is Topological Vector Space, the second is Bourbaki's classic. Other books on Topological Spaces also can be mentioned. However, I want to mention another direction which might interest you.

The Banach Lattice Theory (or more general form), makes it very easy to find Mackey topology. The most important space $L$-space and $M$-space can be identify with $L_1$ and $C(X)$, therefore, it is extremely easy get from weak to Mackey thanks to Banach–Alaoglu theorem , the basic reference for lattice theory is enter link description here. However, it doesn't reveal the full power. For the full power, see this notorious hard-to-read book, from chatper 1 to chapter 10, as well as the appendix.

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