I just want to see the importance of reflexive Banach spaces and what is special about them compared to other Banach spaces. What kind of properties hold in reflexive spaces that do not necessarily hold for a random Banach space. The only property I know is that the closed unit ball of a reflexive space is compact in the weak topology.
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1$\begingroup$ The most commonly used Banach spaces are Hilbert Spaces and $L^p$ spaces, both of which are reflexive. Of course in the case of a Hilbert space, the dual can be identified with the Hilbert space itself, which is a stronger statement. Here we may view an element of a Hilbert space as functional. This is where the bra-ket notation comes in handy in Quantum mechanics. $\endgroup$– JoelAug 4, 2014 at 14:57
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Reflexive spaces interest mathematicians because they have a lot of nice properties:
The unit ball is weakly compact, so you can exploit compactness to prove existence of fixed points, convergent subsequences and etc.
Reflexive spaces are characterized by the property that weak and weak* topology coincide. You can forget about weak topology and work with much-well-understood weak* topology.
Every functional on a reflexive space attains its norm. Simply speaking, you always have a vector that tells you almost everything about your functional. More on this matter you can find here.
Reflexivity is a three-space property. You can pass to quotients and subspaces of reflexive spaces and get a reflexive space again.
After equivalent renorming, all reflexive spaces are strictly convex. In some sense, the unit ball of a reflexive spaces is round.
Reflexive spaces have the Radon–Nikodym property. This allows you to develop a rich theory for vector-valued integration and vector-valued measures for reflexive spaces.
Reflexivity is a rare property and this helps one to distinguish Banach spaces. For example there are no infinite-dimensional reflexive $C^*$-algebras, so $c_0$, $l_\infty$ are not reflexive. Their non-commutative counterparts $\mathcal{K}(H)$ and $\mathcal{B}(H)$ are not reflexive either.
Schauder bases in a reflexive space are very nice and sweet; they are shrinking and boundedly complete. Beware! There are hereditarily indecomposable (and a fortiori without any basis) reflexive Banach spaces. See this paper.
To find more on reflexive spaces use search on this site, or mathoverflow, or any book on Banach geometry with keyword 'reflexive'.
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$\begingroup$ Wow, that's a good list, thank you Norbert !! $\endgroup$ Aug 4, 2014 at 21:21
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$\begingroup$ Thank you. Your answer helped me greatly. I was wondering if you could recommend a book that treats reflexive spaces in more depth than for example Kreyszig? (If you don't know it: Kreyszig is an introductory book into functional analysis. It is very general.) $\endgroup$– user167889Mar 5, 2015 at 5:01
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$\begingroup$ @student There s no book specializing on reflexive spaces. You should look for specific chapters in different books. See this answer for my suggestions. $\endgroup$– NorbertMar 5, 2015 at 10:58