## Continuous linear operators between Banach spaces

Some of my concerns while taking the class Functional Analysis with Professor Vladimir Sverak, who helped me answer these questions:

• The spectral radius may not depend continuouly on the continuous linear operator
• Existence of Fredholm operators between two Banach spaces
• Connectivity of the group of invertible elements of a complex Banach algebra

• ## Linear Partial Diffrential Equations with Constant Coefficients

The write-up 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.

• Linear PDE with constant coefficients

• ## Some remarks on Lie Algebra

The 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.

• On Chapter 1
• On Chapter 2
• Diagonalization and Jordan normal form
• Simultaneous diagonalization of commutative matrices

• ## An Exposition for Leray's paper

Here 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 Dong-Du's paper (2007) for references. The write-up is separated into two parts according to the structure of Leray's paper.

• Part 1: Mild solutions
• Part 2: Leray's weak solutions

• ## Logarithm of Complex Variable

Below 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".

• Define logarithm by power series
• Connection between two derivations

• ## Compact Operators

Below are some remarks I made in Summer 2011 when I learned Compact Operators from a series of online notes by Professor Paul Garrett.

• Compact operators on Hilbert spaces
• Hilbert-Schmidt operators and tensor product of two vector spaces
• Wedge product
• Verify some points in Chapter 3, Lieb and Seiringer-"The Stability of Matter in Quantum Mechanics".