Principles: Uniform Boundedness

by | Aug 23, 2020 | Pure Maths

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

by | Aug 16, 2020 | Problem Solving

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 be unbounded and open. Then let the set     Prove that is dense in...

Principles: Baire’s Lemma

by | Aug 13, 2020 | Pure Maths

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...

Analysis Tricks Part 1

by | Dec 28, 2018 | Problem Solving, Pure Maths

I recently learnt a really neat trick when it comes to proving two quantities are equivalent. It started of with my supervisor claiming that for some and , (1)   is obvious. To me, it was not obvious at all, and may not be so obvious to you either. But what my...

Putnam 2017 A2

by | Dec 16, 2018 | Problem Solving

I am currently in China doing a lot of nice work on the wave equation, can’t wait to tell you all about it! But here is a nice problem I did for fun. Today we will look at a Putnam Problem, here is the question, (screenshot from the problem sheet that can be...

On the definition of a Topological Manifold

by | Dec 6, 2018 | Pure Maths

EDIT 7/1/2019: The proof for each finite subset of a Hausdorff space is closed is corrected. Before I begin, as stated in my articles section, I said I am going to start writing about all the neat maths I’ve learnt throughout my final year project in physics, in...

The beautiful language of tensors

by | Aug 22, 2018 | Physics, Pure Maths

You’ve probably heard the word ‘tensor’ before, but if you’re like me, you’ve gone through your entire physics degree without knowing what they are. So today, I’m going to express the relativistic wave equation in the language of...