Exercise 8, § 1

Prove that in a space which is an infinite product of topological vector spaces (over $\mathbf{R}$ or $\mathbf{C}$) none just the point 0, there does not exist a countable base for the canonical bornology (first show that it is enough to prove this for the space $\mathbf{R}^{\mathbf{N}}$, and then use III, p. 38, exerc. 4).