A non-reflexive Banach space isometric with its second conjugate space.

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For x = (x_{1}, x_{2}, x_{3}, …), let ∥x∥ = sup[∑(x_{p_{i}} − x_{p_{i+1}})^{2} + (x_{p_{n+1}} − x_{p_{1}})^{2}]^{\frac{1}{2}} where supremum is over all positive integers n and all finite increasing sequences of at least two positive integers p_{1}, p_{2}, …, p_{n+1}. Let B be the Banach space of all x for which ∥x∥ is finite and lim_{n} x_{n} = 0. Then B is isometric with B^{##}, but is isometric under natural mapping with a closed maximal linear subspace of B^{##}. This example is due to R.C. James (cf. [Jam]). Ref.

[Jam] R.C.James, A non-reflexive Banach space isometric with its second conjugate space, Proc.of.nat.Acad. of sci., Vol 37, No 3, pp. 174-177, 1951.