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....

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 $u\in A$.

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...