We develop the basics of the theory of infinite dimensional polyhedra in real separable Hilbert spaces. In particular we show that any closed convex set is an s-polyhedron (i.e., a countable intersection of closed semispaces) and that the ensuing set of inequalities can be represented using a continuous linear matrix operators and bound vectors in the Hilbert space. We also study in detail the positive cone of the space, in view of its fundamental role in the theory of polyhedra developed from the range space point of view.