Exercise 7, § 1

Let F be the direct sum space $\mathbf{R}^{(\mathbf{N})}$, G the space $\ell^1(\mathbf{N})$ (I, p. 4, Example); F and G are put in duality by the bilinear form

for $x = (\xi_n) \in F$ and $y = (\eta_n) \in G$. a) Show that, in F, every set K which is convex and compact for σ(F, G) is finite dimensional. (Assume the contrary; let $(n_k)$ be a strictly increasing sequence of integers > 0, and $(a_k)$ a sequence of points of K such that the components of $a_k$ for indices > $n_k$ are zero, but that of index $n_k$ is ≠ 0. Show that there exists a sequence $(t_k)$ of numbers > 0 such that $\sum_k t_k < + \infty$ and that in the Banach space $\mathscr{B}(\mathbf{N})$ of all bounded sequences of real numbers, the point $\sum_k t_k a_k$ has non-zero components for indices $n_i$, for all i; deduce that the sequence of partial sums $s_p = \sum_{k=1}^p t_k a_k$ cannot have a limit point in F for σ(F, G).) b) Show that in F there exist compact sets for σ(F, G) which are infinite dimensional (observe that G is the dual of F for the topology induced by the topology of the normed space $\mathscr{B}(\mathbf{N})$). Show that there exist precompact sets in F for σ(F, G) which are not relatively compact.

c) Deduce from a) and b) that $\tau(G, F) = \sigma(G, F)$, but that $\tau(G, F)$ is distinct from the topology of uniform convergence on subsets of $F$ which are compact for $\sigma(F, G)$.