The open mapping theorem (or the Banach-Schauder theorem, if you prefer) is an incredibly important, relatively straightforward and digestible result in functional analysis which plays a crucial role in a large variety of other interesting theorems. As an exercise, we’ll prove the open mapping theorem here in the standard fashion. This will hopefully serve as a useful reference for later posts about metric regularity and linear openness, which serve as a measuring device to quantify differing degrees of linear openness.
In this post, we’ll talk about the trace parameterization of nonnegative Hermitian trigonometric polynomials, providing a proof which depends on the Kalman-Yakubovich-Popov lemma. This follows very closely along the lines of Chapter 2, Section 2.5 of Dumitrescu’s book “Positive Trigonometric Polynomials and Signal Processing Applications” (Springer, 2007).
As a helpful review, here are a variety of problem and solutions to exercises from an introductory functional analysis class. This is the first installment in a series of functional analysis exercises.