The Orlicz–Pettis theorem for locally convex cones

We present a generalization of the classical Orlicz–Pettis theorem about subseries convergence in topological vector spaces. In preparation we review some aspects of the theory of locally convex cones, a generalization of locally convex topological vector spaces. We introduce conical extensions of the classical sequence spaces and a version of Schur’s theorem about weak and norm convergence of sequences in the extension of \(\ell ^1.\) ${\ell }^1.$ Our version of the Orlicz–Pettis theorem refers to series of bounded elements of a locally convex cone.