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Why are reflexive spaces important intuitively? I know that it brings about many good properties: Properties of reflexive Banach spaces. However, what is so special about reflexive spaces that make these good properties happen? Is there a general intuition of this?

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I am not sure how much intuition there is, when it comes to Banach spaces and why we "like" certain properties more than others except for the reason you mentioned: if we take them away things get more complicated and I guess in some sense less intuitive and less desirable and more pathological.

I guess one way of looking at it would be to recognize that in some sense the "nicest" Banach spaces are the Hilbert spaces. They have many of the important properties we need in order to model and study more complex problems for which the finite dimensional case is simply not "rich enough". (Say more rich structures when it comes to spectra of linear operators and so forth.) However, they also do share a lot with their smaller finite dimensional brothers. They can be canonically identified with its dual, they admit a useful concept of a bases and they are indeed all reflexive.

So maybe one way of thinking about Banach spaces and trying to classify them is to see "how much they differ" from those nice Hilbert spaces. As it turns out non-reflexivity is one of those important properties in order to rescue for instance some topological intuition.

Not sure if that helped. Maybe more like a long comment.

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