Proposition 8

Let P be the set of all periodic continuous mappings of $\mathbf{R}^m$ into $\mathbf{C}$ whose group of periods contains $\mathbf{Z}^m$. Then every function belonging to P can be uniformly approximated in $\mathbf{R}^m$ by linear combinations, with complex coefficients, of functions of the form

where the $h_i$ are rational integers (such linear combinations are called trigonometric polynomials in m variables).

We have only to observe that P (endowed with the topology of uniform convergence) is canonically isomorphic to the space of all continuous mappings of the compact space $\mathbf{T}^m$ into $\mathbf{C}$ (Chapter VII, § 1, no. 6), and apply Proposition 7 to the set of mappings of $\mathbf{T}^m$ into $\mathbf{C}$ which correspond to the m mappings $(x_1, x_2, \ldots, x_m) \to e(x_i) \quad (1 \leq i \leq m)$ of $\mathbf{R}^m$ into $\mathbf{C}$.