Exercise 9, § 4

Let E and F be two vector spaces each with its finest locally convex topology. a) Show that if E and F each have an enumerable basis then every bilinear mapping of $E \times F$ in a locally convex space G is continuous (use Du Bois-Reymond’s th. FVR, V, p. 53, exerc. 8)). b) If one of the spaces E, F has a basis with cardinal equal to that of the continuum, show that there exists a non-continuous bilinear form in $E \times F$. (Reduce to the case where $E = \mathbf{R}^{(\mathbf{N})}$, $F = \mathbf{R}^{\mathbf{N}}$, so that F can be identified with $E^*$ and the bilinear forms on $E \times F$ correspond bijectively with the linear mappings of $E^*$ in itself; then consider the identity mapping of $E^*$, and note that in $\mathbf{R}^{\mathbf{N}}$, a compact set for the product topology cannot be absorbent.)