It's that time of the year. Classes are starting up. You're nervous and excited to be taking some of your first "real" math classes called things like "Abstract Algebra" or "Real Anaylsis" or "Topology." It goes well for the first few weeks as the professor reviews some stuff and gets everyone on the same page.… Continue reading Surviving Upper Division Math

# Category: analysis

## PDE’s and Frobenius Theorem

I've started many blog posts on algebra/algebraic geometry, but they won't get finished and posted for a little while. I've been studying for a test I have to take in a few weeks in differential geometry-esque things. So I'll do a few posts on things that I think are usually considered pretty easy and obvious… Continue reading PDE’s and Frobenius Theorem

## Harmonic Growth as Related to Complex Analytic Growth

Let's change gears a bit. This post will be on something I haven't talked about in probably a year...that's right, analysis. Since the last post was short, I'll do another quick one. The past few days have had varying efforts to solve a problem of the form if $latex f$ is an analytic function and… Continue reading Harmonic Growth as Related to Complex Analytic Growth

## Banach Algebra Homomorphism

I'm in no mood to do something challenging after this last ditch effort to learn analysis before my prelim, so I'll do something nice (functional analytic like I promised) that never ceases to amaze me. Theorem: If $latex \phi$ is a complex homomorphism on a Banach algebra A, then the norm of $latex \phi$, as… Continue reading Banach Algebra Homomorphism

## Not Compact Unit Ball

Here is a beautiful little theorem. The unit ball in an infinite dimensional Hilbert space is not compact. The proof is quite simple. So the unit ball $latex B=\{v\in \mathcal{H} : \|v\|\leq 1\}$. Recall that since this is a Hilbert space, we have an inner product defining this norm $latex \langle v, v \rangle=\|v\|^2$. Since… Continue reading Not Compact Unit Ball

## Fourier Series Theorem

I had many things that I wanted to talk about, but when I read this theorem, it was so shocking that I just had to post it. Now from a general intuition standpoint, you might think this theorem to be quite natural. But remember, most of us have been trained to think Fourier series are… Continue reading Fourier Series Theorem