Continuous linear operators between Banach spacesSome of my concerns while taking the class Functional Analysis with Professor Vladimir Sverak, who helped me answer these questions:
Linear Partial Diffrential Equations with Constant CoefficientsThe writeup was motivated by the question: does the Poisson equation Delta(u) = f have a solution in the whole space R^n provided that f is smooth (yet may not decay at infinity)? The answer is yes. The result is also true for all other linear differential operators with constant coefficients. It was proved in the 1950s by Magrange and Ehrenpreis independently.
Some remarks on Lie AlgebraThe textbook from which I learned the subject is Goodman  Wallach "Symmetry, Representations and Invariants". These remarks were written I took a class with Professor Jiang Dihua in 2014.
An Exposition for Leray's paperHere is an exposition for the famous paper of Leray "On the Motion of a Viscous Liquid Filling Space" (1934). It was written in Summer 2014 after I took the course "Topics in PDE" with Professor Vladimir Sverak. Beside the series of his lecture notes, I used Kato's paper (1984) and DongDu's paper (2007) for references. The writeup is separated into two parts according to the structure of Leray's paper.
Logarithm of Complex VariableBelow are some remarks about the logarithm function I made in Fall 2012 when I learned Complex Analysis from Professor Ben Brubaker and the textbook Ahlfors"Complex Analysis".
Compact OperatorsBelow are some remarks I made in Summer 2011 when I learned Compact Operators from a series of online notes by Professor Paul Garrett.
