"$\vee$ defines an involutory contravariant autofunctor on the category of Hausdorff abelian locally compact topological groups" loool what a way to say $A \cong (A^\vee)^\vee$
@EdwardEvans the nice thing about class formations is that they provide a common framework which is purely algebraic and applies to both local and global CFT (though admittedly, it's harder to check the axioms in the global case), which means that you don't have to do the same things twice, you can just check them once in an abstract setting
Historically global CFT was proved first via L-functions (so you had to use some complex analysis) by Artin and others. Modern algebraic proofs of global CFT actually use local CFT
Lubin-Tate theory is nice because it is very explicit, but I think overall you learn more from the homological approach because group cohomology appears all over ANT
I think we're going from Schneider's book "p-adic Galois reps and $(\varphi, \Gamma)$-modules" and it mentions that Fontaine's original approach to the theory can be replaced by perfectoid magic
Question: if $u=\sum\limits_{i=1}^\infty\langle u,e_i\rangle e_i$, then does $\overline{u}=\sum\limits_{i=1}^\infty\langle u,e_i\rangle \overline{e}_i$? ($e_i$ constitute an orthonormal basis)
I'm thinking yes, but my "geometric intuition" has been wrong before
a general complex vector space doesn't have a natural complex conjugation that is independent of choosing a basis (unless it's a function space or something)
@Rithaniel I don't think you need to work with an orthonormal basis for that problem. Just manipulate the inner products a little bit: $\langle \langle x,u\rangle v,w\rangle=\langle x, u\rangle\langle v,w\rangle=\langle x,\overline{\langle v,w\rangle}u\rangle=\langle x,\langle w,v\rangle u\rangle$ this shows that $w \mapsto \langle w,v\rangle u$ is the adjoint
Is there a functional analytic/fourier analytic version of the central limit theorem? Something along the lines of "if I convolute a pdf often enough, the corresponding sequence of integral operators defined on the bounded, continuous functions converges weakly to the integral operator corresponding to the normal distribution"
By these integral operators I just mean the expectations
there's a measure-theoretic version of the CLT stating that the if you convolve a properly normalized Borel probability measures with itself subsequently and rescale, the resulting sequence of measures weakly converges to the standard Gaussian measure
@AlessandroCodenotti I once took a whole homotopy type theory course where the highlight was me not understanding how the equality type in the simplicial model actually works
there's an extension of CLT to polynomially decaying random variables for which the scaled limits converge to stable $\alpha$-symmetric distributions. $\alpha = 1/2$ is the Gaussian
@TedShifrn Have I told you that if $B$ is symmetric idempotent, $Q$ is symmetric positive definite such that $I - B - Q$ is positive definite, then $BQ = QB = 0$
I will give you a very natural, category theoretical introduction
If $X$ is a random variable, we can consider random variables of the form $X' A X$ where $A$ is some matrix. These are the so called "randomized quadratic forms", where you take a quadratic form and feed sample values from a specific law $X$ to it, and you get a new random variable/measure out of it.
Simplest example is $X \sim N_n(\mathbf{0}, I_n)$, multivariable standard normal, and you look at $X' X$. This is the $\chi^2_n$ distribution.
