Suppose that P is a cone of a normed space X. A vector \(x_0\in P\) is a semi-interior point of P if a real number \(\rho > 0\) exists so that \(x_0-\rho U_+\subseteq P\) , where \(U_+=P\cap U\) is the positive part of the unit ball U of X. In this article we extent the notion of semi-interior point in topological vector spaces X ordered by a wedge P with a detailed study of these points and properties of the spaces with semi-interior points. To this end we define a new locally convex topology, the \({\mathcal {P}}\) -topology of X, in which the family of sets \(co(U_+\cup (-U_+))\) , where U is a circled neighborhood of zero, is a fundamental system for this topology. We show that if P has semi-interior points, the topological dual \((X,{\mathcal {P}})^*\) of \((X,{\mathcal {P}})\) and the order dual \(X^\thicksim \) of X, coincide. Also we show an analogous result for a space of operators. Finally note that semi-interior points have applications in mathematical economics and also in E-metric spaces.
