Exercise 23, § 1

Give an example of a family $(E_i)_{i \in I}$ of Hausdorff locally convex spaces such that the canonical mapping from $\bigoplus_{i \in I} E'_i$ onto the dual of $P = \prod_{i \in I} E_i$ is not an isomorphism from the topological direct sum of the $E'_i$ endowed with the weak topologies $\sigma(E'_i, E_i)$ onto the dual $P'$ endowed with $\sigma(P', P)$.