Exercise 24, § 5

Let X be the interval [0,1] of $\mathbf{R}$, μ the Lebesgue measure on X. Show that for every Banach space F, every continuous linear form on $\mathscr{S}_F$ is identically zero. (For every neighborhood V of 0 in $\mathscr{S}_F$, show that there exists an integer n > 0 such that V + V + ... + V (n terms) contains a line.)