User:Caliburn/Job List

Real Analysis

 * Results about divergent sequences. The result that if $\sequence {x_n}$ is a sequence with $x_n \to +\infty$ and $\sequence {x_{n_j} }$ is a subsequence of $\sequence {x_n}$ then $x_{n_j} \to +\infty$ too was missing and was put here: Subsequence of Real Sequence Diverging to Positive Infinity Diverges to Positive Infinity. Essentially I'm just throwing these up as intermediate results, someone needs to go through and give a proper treatment, and link back to extended real numbers at some point.


 * Results on common limits of functions. We have Exponential Dominates Polynomial, but that's about it. When I wrote Order of Natural Logarithm Function, I was surprised we didn't have a page on the limit of $x^\alpha \paren {\ln x}^\beta$. We don't have a category pertaining to limits at infinity.


 * Proper treatment of indefinite integrals. Do we want to talk about indefinite integrals as cosets in a particular function quotient space? This treatment is rare since indefinite integrals disappear before courses on real analysis.


 * Multivariable analysis - many gaps. To name a few, directional derivative, Frechet derivative, derivatives of functions $\R^n \to \R^m$, interplay with Jacobian, derivative of matrix functions (including $A \mapsto A^{-1}$ and $A \mapsto A^n$, $n \ge 0$), chain rules, inverse function theorem, implicit function theorem, submanifolds of $\R^n$.

Advanced

 * Rescue pages on the $\delta$ "function" and re-interpret things in terms of distributions. It is worth mentioning abuses of notation present in engineering and physics.


 * More work on distribution theory piggybacking off the above.


 * Interplay between distributional/weak derivatives and classical derivatives


 * Sobolev spaces


 * Bochner spaces


 * The Definition:Cauchy Principal Value is set up, but isn't used anywhere. Applications include the Hilbert transform, solving the distributional equation $T \cdot x = 1$ and so on.

Measure Theory

 * A full treatment on the extended real numbers, including limits in $\overline \R$, Bolzano-Weierstrass in $\overline \R$, etc.


 * $L^p$ spaces are not properly set up. This includes arithmetic (which is "obvious" addition, pointwise multiplication, scalar multiplication, reciprocal of equivalence classes, and showing these are well-defined) and integration.


 * Duality in $L^p$ spaces


 * Most interchanges of sums/limits, integrals/limits, sums/sums, etc. are not properly justified. (I was guilty of handwaving this years ago) The site isn't quite ready for this - first we will need to cover the interplay between Lebesgue and Riemann integration (on my to-do list), and translate the Monotone Convergence Theorem, Dominated Convergence Theorem, Fubini/Tonelli's Theorem into appropriate results for the Riemann and Lebesgue integral. Then it'd be a matter of going through Special:WhatLinksHere/Fubini's Theorem, Special:WhatLinksHere/Tonelli's Theorem, and so on and fixing the instances. If the technical details are unwieldly and distract from the flow of the page, include this as a lemma.


 * Some results on indefinite integrals involve interchanges, so they would have to be changed to definite integrals leaving one bound arbitrary.

Functional Analysis

 * Many definitions are unnecessarily specific, referring to Hilbert spaces instead of general inner product spaces, Banach spaces or normed vector spaces. I've started some of them in my sandbox. Feel free to finish them and bring them into definitionspace.


 * Flesh out/tighten stuff, I will try to cover this stuff over the summer:


 * Hahn-Banach Theorem - proof seems weak/incomplete


 * Second dual, reflexivity - nothing here yet it looks like


 * Weak convergence, Banach-Alaoglu Theorem - fill in redlinks


 * Spectral theory - absent

Analytic Number Theory

 * Build towards rewriting Prime Number Theorem. Tried it and got bored, sorry