Example

— 1) Let $\ell^\infty(\mathbf{N})$ denote the Banach space of bounded sequences $x = (x_n)_{n \in \mathbf{N}}$ of scalars, with the norm

(3)

Let $c_0(\mathbf{N})$ be the closed vector subspace of $\ell^\infty(\mathbf{N})$ consisting of sequences tending to 0. Finally, let $\ell^1(\mathbf{N})$ be the vector space of summable sequences, endowed with the norm

(4)

We can show (IV, p. 47, exerc. 1) that the dual of $c_0(\mathbf{N})$ can be identified with $\ell^1(\mathbf{N})$ in such a way that we have

(5)

for all $\mathbf{x} \in c_0(\mathbf{N})$ and $\mathbf{x}' \in \ell^1(\mathbf{N})$. Similarly the dual of $\ell^1(\mathbf{N})$ can be identified with $\ell^\infty(\mathbf{N})$ in such a way that we have the relation (5) for all $\mathbf{x} \in \ell^1(\mathbf{N})$ and all $\mathbf{x}' \in \ell^\infty(\mathbf{N})$. Hence $\ell^\infty(\mathbf{N})$ is the bidual of $c_0(\mathbf{N})$, and this latter space is not reflexive.

\* 2) Every hilbertian space is a reflexive Banach space (V, p. 17). \* 3) Let X be a Hausdorff topological space and $\mu$ a complex measure on X. For every real number $p > 1$, the Banach space $L^p(X, \mu)$ is reflexive, and its dual can be identified with $L^q(X, \mu)$ with $p^{-1} + q^{-1} = 1$ (INT, V, 2nd edition, § 5, No. 8 and IX, § 1, No. 10).