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Measure theory and $L^p$ Spaces

I'm on a quest to learn about operator algebras in the hopes of understanding the many interesting ways they have been applied to quantum field theory. This note is the first in a series that will build the essential aspects of the theory from the ground up. I will not prove everything, giving references for the proofs of many lemmas, but I will try to give enough detail that the mathematical underpinnings of the theory are clear. This post is about $L^p$ spaces. These are Banach spaces of functions — with the $L^{\infty}$ space actually being a Banach algebra — that show up frequently in the theory of general operator algebras. I will introduce the basics of measure theory and the general theory of Lebesgue integration, introduce the $L^p$ spaces, and prove some basic theorems concerning them. The outline is: In section 1 , I will introduce the basic tools of measure theory: $\sigma$-algebras, measurable functions, and measures. In section 2 , I will discuss some elementary proper

Ordinal numbers, transfinite induction, and Zorn's Lemma

Hello from Cambridge, Massachusetts! I've just arrived this past week from California, as August 31 marks my last day as a PhD student at Stanford and September 1 my first day as a postdoctoral researcher at MIT. I've been on a significantly reduced work schedule during the move, so rather than working directly on research problems, I've been taking time to bone up some fundamental math skills. This has involved, mostly, an exploration into functional analysis. One of the fundamental theorems of functional analysis is the Hahn-Banach theorem , which is proved using a very standard application of Zorn's lemma . During my reading, though, I realized that I actually had no idea how Zorn's lemma is proved. I've applied it many times in math courses, but the underpinnings of the lemma elude me. Trying to figure out the proof led me down a pretty deep rabbit hole — it turns out Zorn's lemma is basically a simplified packaging of a general technique called "tr

Hilbert spaces of Majorana fermions

When we talk about fermions in quantum mechanics, we talk about two kinds: Dirac and Majorana. Both of these are supposed to have the property that if I create a fermion of type $1$ and then a fermion of type $2,$ the resulting state will be related by a minus sign to the state where I create a fermion of type $2$ and then a fermion of type $1.$ But there is a decision to make as to what should happen if we try to create two type-$1$ fermions. Dirac fermions are defined by the property that if you try to create two type-$1$ fermions, the state is completely annihilated to the zero vector. Majorana fermions are defined by the property that if you try to create two type-$1$ fermions, they annihilate one another and leave the total state unchanged. These properties are summarized by saying that the algebras of operators that create and annihilate the two different types of fermions should be different. The Dirac fermions have creation and annihilation operators $a_j, a_j^{\dagger}$ satisf