Study Irreducible Representations of SU(2) Using Fourier Series
$SU(2)$ has a lot of interesting mathematical and physical properties. In this post we study its irreducible representations in a mathematician's way.
The Group Algebra of A Finite Group and Maschke's Theorem
In this post we give several forms of Masher's theorem by studying group algebra, which eventually becomes a study of semisimple rings. One can consider this post a chaotic evil introduction to representation theory or something.
The ring of trigonometric polynomials with complex scalars
This post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.
The ring of real trigonometric polynomials
The ringThroughout we consider the polynomial ring R=\mathbb{R}[\cos{x},\sin{x}].This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.
The concept of generalised functions (distributions) and derivatives
In this post, we study the concept of generalised functions (a.k.a. distributions), and let's see how to evaluate the derivative no matter the function is differentiable or not.
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...