Question

In: Advanced Math

definitions of weak topology and weak star topology with as little jargon as possible.

Expert Solution

Let be a non empty set and be a family of topological spaces. Let . Then we define weak topology on in such a way that each is continuous with respect to in the following way:

Since we expect each to be continuous with respect to , if , then . For to be a topology, it should be closed under arbitrary unions. Let , then the topology generated by is the required topology, i.e., is the basis of the weak topology generated by .

Let be a normed space and be its dual space. For each , the functional , is a semi-norm on . The topology induced by the family of seminorms is the weak topology on . Thus, weak star topology is nothing but the weak topology generated on the dual space by the family of functions which will be continuous in this topology.

The aim to define this kind of topologies is to simplify our study of the space of continuous functions because every function on the topological space (with weak topology) is continuous.

Colby Messinger answered 1 year ago

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