This course had a significant influence on my point of view on lot’s of different problems, from study of different ODE/PDE’s to even various kinds of “Optimization” problems, because technically all of those concepts (i.e. ODE/PDE’s or Optimization problems) are certain operators from one space to another. Below is details of what we covered in that course:
During this course we first had a review of Hilbert and Banach Spaces, then we fully studied different kinds of operators such as “Linear”, “Bounded”, “Compact”, “Hermitian”, “Self-Adjoint” and “Unitary” Operators. In the second part of the course we reviewed several versions of “Spectral Theorem” with proofs for it.
In the last part of the course we first studied several kinds of unbounded operators such as “Closed and Closeable, Symmetric and Self-Adjoint” unbounded operators and after that we learned Spectral and Bloch Theory for unbounded operators and the spectrum of periodic Schrodinger operators.