Example

— Let E be the Banach space (over $\mathbf{C}$) consisting of continuous complex functions with period 1 in $\mathbf{R}$, with the norm $\|f\| = \sup_x |f(x)|$.

For every integer $n \in \mathbf{Z}$ and every function $f \in E$, let $c_n(f) = \int_0^1 f(x) e^{-2i\pi nx} dx$ ($n$-th Fourier coefficient of $f$) ; each of the mappings $f \mapsto c_n(f)$ is a continuous linear form on E. Let $(\alpha_n)$ be a sequence of complex numbers such that, for every function $f \in E$, the serie with the general term $\alpha_n c_n(f) + \alpha_{-n} c_{-n}(f)$ is convergent. Under these conditions, the mapping $u : f \mapsto \alpha_0 c_0(f) + \sum_{n \geq 1} [\alpha_n c_n(f) + \alpha_{-n} c_{-n}(f)]$ is a continuous linear form on E ; \* in other words, there exists a measure $\mu$ on $[0, 1]$ such that $u(f) = \int f(x) d\mu(x)$ for every function $f \in E$, and $\alpha_n$ is the $n$-th Fourier coefficient of $\mu$.* In fact, for every integer $m > 0$, the mapping $f \mapsto \sum_{k = -m}^m \alpha_k c_k(f)$ is a continuous linear form $u_m$ on E, and for all $f \in E$, the sequence $(u_m(f))$ converges to $u(f)$, by hypothesis. The assertion then follows from Banach-Steinhaus theorem, since E is barrelled.