The Riesz-Markov-Kakutani Representation Theorem
In this post we develop a proof of the Riesz-Markov-Kakutani theorem on a locally compact Hausdorff space, which is the essential of the existence of the Lebesgue measure.
The Big Three Pt. 4 - The Open Mapping Theorem (F-Space)
The Open Mapping TheoremWe are finally going to prove the open mapping theorem in $F$-space. In this version, only metric and...
The completeness of the quotient space (topological vector space)
The GoalWe are going to show the completeness of $X/N$ where $X$ is a TVS and $N$ a closed subspace. Alongside, a bunch of us...
An Introduction to Quotient Space
I’m assuming the reader has some abstract algebra and functional analysis background. You may have learned this already in yo...
The Big Three Pt. 3 - The Open Mapping Theorem (Banach Space)
What is open mappingAn open map is a function between two topological spaces that maps open sets to open sets. Precisely spea...
A Brief Introduction to Fréchet Derivative
In this post we give a brief introduction too Fréchet derivatives, a generalisation of derivatives in Banach spaces, and deduce *elementary* properties.
The Big Three Pt. 2 - The Banach-Steinhaus Theorem
About this blog postPeople call the Banach-Steinhaus theorem the first of the big three, which sits at the foundation of line...
The Big Three Pt. 1 - Baire Category Theorem Explained
About the ‘Big Three’There are three theorems about Banach spaces that occur frequently in the crux of functional analysis, w...