Topics
Below is an outline of the contents of this course. Note that we might not have time to cover everything.
Preliminaries
- Basic properties of real numbers
- Elementary set theory
- Functions
- Cardinality
- Supremum and infimum of sets
- Vector spaces
- Span, independence and dimension
Metric spaces
- Metrics
- Cauchy sequences
- Convergence and completeness in metric spaces
- Open and closed sets, neigborhoods, bounded sets, accumulation points and boundary points
- Banach's Fixed Point Theorem and applications thereof
Norms and Banach spaces
- Norms and normed spaces
- The \(\ell^p\) spaces
- Induced metric
- Banach spaces
- Equivalent norms
- Span, closed span and complete sequences
- Weierstrass Approximation Theorem
Inner products and Hilbert spaces
- Inner products
- Orthogonal and orthonormal sets and sequences
- Orthogonal projection and the Closest Point Theorem
- Bessel's inequality
- Orthonormal bases
- Fourier coefficients
- Plancherel's and Parseval's identity
Operator Theory
- Linear operators on normed spaces
- Bounded operators
- The space \(B(X,Y)\)
- Dual spaces
- Riesz Representation Theorem
- Unitary operators
- Adjoint (and self-adjoint) operators
- The Spectral Theorem for self-adjoint operators
Linear transformations on finite-dimensional vector spaces
- Matrices
- Rank and rank-nullity theorem
- Eigenvalues and eigenvectors
- Similar matrices
- Jordan normal form (and applications)
- LU decomposition
- Singular value decomposition (SVD)