Stirling公式的几种经典估计
Stirling公式对于$\Gamma$函数，我们有一个经典的极限式（证明请见ProofWiki）。 \lim_{n\to\infty}\frac{\Gamma(x+1)}{(x/e)^x\sqrt{2\pi{x}}}=1.利用这个式子，我们能立刻...
A proof of the ordinary Gleason-Kahane-Żelazko theorem for complex functionals
The Theorem (Gleason-Kahane-Żelazko) If $\phi$ is a complex linear functional on a unitary Banach algebra $A$, such that $\ph...
The Big Three Pt. 5 - The Hahn-Banach Theorem (Dominated Extension)
About this postThe Hahn-Banach theorem has been a central tool for functional analysis and therefore enjoys a wide variety, m...
A long exact sequence of cohomology groups (zig-zag and diagram-chasing)
Exterior differentiation(This section is intended to introduce the background. Feel free to skip if you already know exterior...
The Fourier transform of sinx/x and (sinx/x)^2 and more
In this post we compute the Fourier transform of $\sin{x}/x$ and $(\sin{x}/x)^2$ through contour integration.
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...
Introducing Riemann-Stieltjes Integral
MotivationRiemann-Stieltjes integral is a generalisation of Riemann integral, the one every college student studies in their ...
Basic Facts of Semicontinuous Functions
ContinuityWe are restricting ourselves into $\mathbb{R}$ endowed with normal topology. Recall that a function is continuous i...