Reflexive space

Remarks

Since $J$ is a linear isometry, $X$ is reflexive exactly when $J$ is an isometric isomorphism of $X$ onto $X^{**}$. A reflexive space is complete, because $X^{**}$ is complete. Example: for $1 < p < \infty$ and $1/p + 1/q = 1$, the duals satisfy $(\ell^p)^* = \ell^q$ and $(\ell^q)^* = \ell^p$; under these identifications $J$ is the identity of $\ell^p$, so $\ell^p$ is reflexive. Every Hilbert space is reflexive by the Riesz representation theorem. Non-example: $c_0$ is not reflexive, since $J$ is the inclusion $c_0 \subset \ell^\infty$, which misses the constant sequence $(1, 1, \ldots)$. The spaces $\ell^1$, $\ell^\infty$ and $C[0, 1]$ are not reflexive either. The definition asks for the specific map $J$ to be onto: the James space is isometrically isomorphic to its bidual by some other map, but it is not reflexive.