Properties of reflexive Banach spaces

Solution 1:

Reflexive spaces interest mathematicians because they have a lot of nice properties:

Unit ball is weakly compact, so you can exploit compactness to prove exitence 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 unit ball of a reflexive spaces is round.

Reflexive spaces have Radon-Nykodim 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 is 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.

Shauder 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'.