Monthly Archives: May 2012

Hilbert Spaces – Part 2

Posted on 05/28/2012 by Ryan

Recall that in the finite dimensional world, two vector spaces are isometrically isomorphic if and only if they have the same dimension. In the last post, I mentioned that, by using the appropriate definition of a basis and dimension, one … Continue reading →

Posted in Analysis, Linear Algebra | 1 Comment

Littlewood’s Three Principles (3/3)

Posted on 05/26/2012 by Adam Azzam

So far we’ve proven Littlewood’s first and third principles. In this last installment, we’ll prove his second principle, that “every function is nearly continuous.” This result is better known as Lusin’s Theorem.

Posted in Analysis | 2 Comments

Littlewood’s Three Principles (2/3)

Posted on 05/23/2012 by Adam Azzam

Littlewood’s three principles provide useful intuition for those first learning measure theory: “The extent of knowledge [of real analysis] required is nothing like as great as is sometimes supposed. There are three principles, roughly expressible in the following terms: Every … Continue reading →

Posted in Analysis | 2 Comments

Littlewood’s Three Principles (1/3)

Posted on 05/21/2012 by Adam Azzam

When I first studied measure theory, I felt like I could never quite wrap my head around what a measurable set or function looked like. After all, the Borel Heirarchy is huge, and what the hell does a set look … Continue reading →

Posted in Analysis | 4 Comments