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...
Basic Facts of Semicontinuous Functions
ContinuityWe are restricting ourselves into $\mathbb{R}$ endowed with normal topology. Recall that a function is continuous i...
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...
More properties of zeros of an entire function
What’s going on againIn this post we discussed the topological properties of the zero points of an entire nonzero function, o...
The Lebesgue-Radon-Nikodym theorem and how von Neumann proved it
An introductionIf one wants to learn the fundamental theorem of Calculus in the sense of Lebesgue integral, properties of mea...