Weak Topology - The Weak-* Topology

The Weak-* Topology

A space X can be embedded into the double dual X** by

where

Thus T : X → X** is an injective linear mapping, though not necessarily surjective (spaces for which this canonical embedding is surjective are called reflexive). The weak-* topology on X* is the weak topology induced by the image of T: T(X) ⊂ X**. In other words, it is the coarsest topology such that the maps T_x from X* to the base field R or C remain continuous.