The Amazing Baire Category Theorem

One of the most fundamental theorems in mathematics is the famous Baire category theorem, which never ceases to amaze me. If you are a modern analysis enthusiast like me you probably know the feeling when you are working on a seemingly difficult problem and the solution appears to elude you every time you think about it, and then all of a sudden, snap! “Oh, I can just use the Baire category theorem.”

This post is dedicated to this feeling. … More The Amazing Baire Category Theorem

On the nonemptiness of the spectrum

This time we shall explore a topic which really fits the name of the blog. It it a well known and elementary fact that every complex matrix has an eigenvalue and an eigenvector. But, if we think of the matrix as a linear transformation between finite dimensional vector spaces, then what about the spectrum for linear transformations acting on infinite dimensional spaces? The answer for this question is interesting and it utilizes a very beautiful extension of complex analysis for vector valued functions. … More On the nonemptiness of the spectrum