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

## Zeros of Analytic Functions

A strange property of analytic functions is that the zeros are isolated. I don't remember the proof I originally learned of this fact, but today I saw a really interesting topological way to do it. It makes sense now. More precise formulation: If $latex \Omega\subset\mathbb{C}$ is a connected open set, then $latex \{z: f(z)=0\}$ consists… Continue reading Zeros of Analytic Functions

## 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

## Applying Covering Theorems

I've searched far and wide to not do one of the standard applications that are in all grad analysis texts (yes I'm referring to the Hardy-Littlewood maximal function being weakly bounded). We are getting into the parts of analysis that I despise (it will all be over in 4 days...I hope *crosses fingers*). Claim: If… Continue reading Applying Covering Theorems

## Covering Theorem (we use past Lemmas)

A brief break occured while I moved 2700 miles away. The important thing is I'm back, and we're going to prove a big one today. First let's define a Vitali covering. A set is Vitali covered by the collection of sets $latex \mathcal{V}$ if for any $latex \epsilon>0$ and any x in the set, there… Continue reading Covering Theorem (we use past Lemmas)