Several ways to prove Hardy's inequality
Suppose $1 < p < \infty$ and $f \in L^p((0,\infty))$ (with respect to Lebesgue measure of course) is a nonnegative func...
Tensor Product as a Universal Object (Category Theory & Module Theory)
IntroductionIt is quite often to see direct sum or direct product of groups, modules, vector spaces. Indeed, for modules over...
Why Does a Vector Space Have a Basis (Module Theory)
Module and vector spaceFirst we recall some backgrounds. Suppose $A$ is a ring with multiplicative identity $1_A$. A left mod...
Rings of Fractions and Localisation
Is perhaps the most important technical tools in commutative algebra. In this post we are covering definitions and simple pro...
The Grothendienck Group
Free groupLet $A$ be an abelian group. Let $(e_i)_{i \in I}$ be a family of elements of $A$. We say that this family is a bas...
Study Vector Bundle in a Relatively Harder Way - Tangent Bundle
Tangent line and tangent surface as vector spacesWe begin our study by some elementary Calculus. Now we have the function $f(...
A Continuous Function Sending L^p Functions to L^1
Throughout, let $(X,\mathfrak{M},\mu)$ be a measure space where $\mu$ is positive. The questionIf $f$ is of $L^p(\mu)$, which...
Study Vector Bundle in a Relatively Harder Way - Definition
MotivationDirection is a considerable thing. For example take a look at this picture (by David Gunderman): The position of t...
The Big Three Pt. 6 - Closed Graph Theorem with Applications
(Before everything: elementary background in topology and vector spaces, in particular Banach spaces, is assumed.) A surprisi...
Partition of Unity on Different Manifolds (Part 1. Introduction)
An application of partition of unityPartition of unity builds a bridge between local properties and global properties. A nice...