Baire Category Theorem and Its Applications
Exposition of the Baire Category Theorem (equivalent formulations) with applications to the Uniform Boundedness Principle and the Closed Graph Theorem; worked examples include a generic-but-measure-zero set and Grothendieck’s theorem (closed subspaces of L^p contained in L^\infty on finite-measure spaces are finite-dimensional).
Overview
This note gives an exposition of the Baire Category Theorem (via equivalent formulations) and applies it to the Uniform Boundedness Principle and the Closed Graph Theorem.
Worked examples include:
- a generic-but-measure-zero set, and
- Grothendieck’s theorem: closed subspaces of $L^p$ contained in $L^\infty$ (on finite-measure spaces) are finite-dimensional.