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## Solution to Problem 3

So this problem comes from the functional analysis course I took last semester. Although simple, I thought was pretty cool. In the actual exercise they had broken it down into three parts, which I did not show in the problem post. Here I will show the steps as claims....

## Principles: Open Mapping Theorem

So guess what happened again, another WordPress malfunction… something really isn’t right here. The blog post is attached! Open Mapping BlogDownload

## Problem of the Week 3

Supposed that $u, f\in L^2(\Omega)$ satisfies $$\int_{\Omega}u\Delta^2 \phi^2 = \int_{\Omega} f\phi dx,$$ for all $\phi \in A = \{u\in H^4\cap H^1_0 (\Omega) : ~\Delta u \in H^1_0(\Omega)\}$. Prove that $u\in A$.

## Solution to Problem of the Week 2

So once again, WordPress doesn’t like my Latex, so I have attached this blog as a pdf. Enjoy! PrbWk2Download

## Solution to Problem of the Week 1

So for some reason my usual way of writing Latex on the blog doesn’t work for this post. I have no idea why unfortunately, so I will just add the post as a PDF attached to this! Solution to Problem 1Download

## Principles: Uniform Boundedness

So last week we talked about the Baire Category Theorem, and this important tool will give us three equivalent theorems about the qualitative properties of operators on Banach spaces. Today we will begin with the first theorem, uniform boundedness. The proof of this...

## Problem of the Week 1

So this problem is in line with the post of this week, Baire’s Lemma. Have a go and let me know your thoughts in the comments! Let $D\subset (0,\infty)$ be unbounded and open. Then let the set G=\{x\in (0,\infty); nx\in D ~\text{for infinitely...

## Principles: Baire’s Lemma

So today I am starting a new series on the Principles of Functional Analysis to give you all an appreciation for the power that these simple truths hold. We will start today with Baire’s Lemma , which will require some definitions, hopefully, all familiar. (If...