In §1 we show that the well-known "Fichera existence principle in Banach spaces" (See [3], pp. 174-178; [2], [4], pp. 30-50; [5], pp. 11-16) holds—suitably enonciated-in locally convex Hausdorff spaces: the (very simple) proof is founded on the remark that the existence problem is equivalent to a property of inclusion of dual spaces. In §2 we explain how this principle includes, as particular cases, the so-called "Theorem of surjection of Fréchet spaces" (see [7], Theor. 37.2) and a generalisation of it—recently proved (with different method) by G. Zampieri [8]—which are fundamental in some topics concerning the theory of linear partial differential operators.