Approach Theory Meets Functional Analysis

Abstract

Here we start in the classical setting of normed spaces and see how the well-known weak and weak* topologies can be quantified by canonical approach structures and how using index analysis, this allows to obtain quantified results of which several classical results are simple corollaries. In a second part we see that the construction of the weak and weak* approach structures on normed spaces fits into the wider picture of what we call approach vector spaces and locally convex approach spaces.

In most sciences one generation tears down what another has built ... In mathematics alone each generation builds a new story to the old structure.

(Hermann Hankel)