Functional Analysis (4): Dual Spaces and the Hahn-Banach Theorem — Taming Linear Functionals

Thu, 07 Oct 2021 09:00:00 +0000

Why You Cannot Skip This Article#

Up to now, the theory has been about spaces and the elements that live in them. This article changes the perspective: it asks what you can say about a vector $$x$$ by measuring $$x$$ against a family of test functionals. The shift from “vectors” to “vectors plus functionals” is what turns Banach spaces into a serviceable analogue of finite-dimensional linear algebra. In finite dimensions, every linear functional is continuous and the dual space is the same dimension as the original — so there is nothing to prove. In infinite dimensions, continuity is a real constraint, and the existence of enough continuous functionals to separate points or extend partial data is not obvious. The Hahn-Banach theorem is what guarantees this, and it is the result that makes functional analysis possible.

Hahn-Banach on Chen Kai Blog

Functional Analysis (4): Dual Spaces and the Hahn-Banach Theorem — Taming Linear Functionals

Why You Cannot Skip This Article#