Weak Topology

This chapter defines the weak topology of a separated semi-normed space and characterizes weakly convergent sequences, that is, convergent for the weak topology. It shows that the weak topology of a Hilbert space can be expressed in terms of the scalar product and also shows that any continuous linear mapping is weakly continuous, which is a Banach-Dieudonne theorem. The weak topology is preserved by topological equalities. The chapter explains that the weak topology of a product of semi-normed spaces coincides with the topology of the product of the spaces endowed with their weak topologies. It compares the weak topology of an intersection of semi-normed spaces with the topology of the intersection of spaces endowed with their weak topologies. The chapter shows that, in a Hilbert space, the weak convergence jointly with the convergence of the norm implies strong convergence.