Exercise 1, § 3

Let E be a Hilbert space with a countable orthonormal basis $(\mathbf{e}_n )$ (TVS, V, §2, No.3), $E_0$ the dense linear subspace of E formed by the linear combinations of finitely many vectors $\mathbf{e}_n$. Let X be the compact subspace of $\mathbf{R}$ formed by 0 and the points $1/n$ ($n$ an integer $\geqslant 1$); let $\mu$ be the positive measure on X defined by the mass $1/n^2$ at each point $1/n$ and the mass 0 at the point 0 ($§ 1$, No. 3, Example I). Consider the continuous mapping f of X into $E_0$ defined by $f(0) = 0,\ f(1/n) = \mathbf{e}_n / n$ for $n \geqslant 1$; show that the integral $\int f\ d\mu$ does not belong to $E_0$.