I need to know if a certain Banach space I stumbled upon is reflexive or not. I need to know what are the state of the art techniques to determine if a Banach space is reflexive or not. For example, how does one prove that the space of bounded continuous functions is not reflexive. Same question for Linfinity. A description of the methods involved or a reference would both be appreciated.
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A good reference for functional analysis in general is the book by John B. Conway. This book states that if $X$ is a compact space(or more generally a completely regular space), then $C(X)$ is reflexive if and only if $X$ is finite(p. 90). See This link for proofs or you can easily construct an element in $C(X)^{**}\setminus C(X)$ using the Riesz representation theorem. In particular, by Gelfand duality, every infinite dimensional commutative $C^{*}$algebra is nonreflexive, so the space $L^{\infty}(\mu)$ is reflexive if and only if $L^{\infty}(\mu)$ is finite dimensional. 

