Category Archives: Analysis

To Converge or Not to Converge – Hilbert Spaces Part 4/Banach Spaces 1

Posted on 11/04/2012 by Ryan

Last time I introduced the notion of the weak topology on a Banach space . I defined the topology in terms of net convergence, but remarked that this was equivalent to the weak topology generated by . The purpose of … Continue reading →

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Moving Around a Sphere – Hilbert Spaces Part 3

Posted on 10/27/2012 by Ryan

Well, it’s been a long time since I’ve written a post, but I have a bit of time tonight and I really don’t feel like doing homework tonight, so let’s talk about some mathematics! Last time I wrote, I promised … Continue reading →

Posted in Analysis, Set theory | Leave a comment

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

A Probability “Paradox” as per the Previous Post

Posted on 04/07/2012 by Jay Cummings

Last time I mentioned the paradoxical-sounding situation in which I said “pick a number at random under some given continuous probability distribution over the reals, and call this number ”.  But what was the probability of choosing ?  Well, a … Continue reading →

Posted in Algorithms, Probability | 1 Comment

Digression: Hilbert Spaces – Part 1

Posted on 03/15/2012 by Ryan

Hello all, I hope your semesters are going well! I am on spring break this week, so I’ve got a bit of time, and I figured instead of getting a head start on my homework for next week, I’d write … Continue reading →

Posted in Analysis, Linear Algebra | 1 Comment

The Riesz Representation Theorem – Part 2

Posted on 02/28/2012 by Ryan

So I should explain: this is one of those weird weeks where I have actually finished all my homework for the week two days early, so I actually have some time to write on the blog! So, let’s talk about … Continue reading →

Posted in Analysis, Linear Algebra | 3 Comments

The Riesz Representation Theorem – Part 1

Posted on 02/27/2012 by Ryan

Hello everyone! I realize it’s been far too long since I’ve last posted, and I decided I don’t really want to write about Radon-Nikodym anymore. Maybe someday if I get requests I’ll write a couple more posts in that series, … Continue reading →

Posted in Analysis, Linear Algebra | 3 Comments