Topics
Metric and normed spaces
- Metric spaces
- Limits
- Open and closed sets
- Examples of metric and normed spaces: \(\mathbb{R}^n\); sequence spaces \(\ell_1\), \(\ell_2\) and bounded sequences; function space \(C(\mathbb{R})\) with different norms
- Bolzano-Weierstrass theorem
- Cauchy sequences, completeness
Relevant exam problems: 2003-1, 2005-1, 2007-6, 2012-6, 2013-3
Banach fixed point theorem and its applications
- Contractions in metric spaces
- Banach fixed point theorem in a complete metric space
- Proof of the fixed point theorem, iterations
- Applications to functional equations, linear algebra, integral equations
- Lipschitz functions in metric spaces
- Picard-Lindelöf theorem for initial value problem for ODE
- Picard iterations
Relevant exam problems: 2004-2, 2005-4, 2006-1, 2010-2
Vectors spaces, normed spaces
- Vector spaces and subspaces
- Finite basis and dimension
- Norm, examples
- Equivalent norms, all norms in a finite dimensional space are equivalent
- Banach spaces
Relevant exam problems: 2004-7, 2012-4, 2005-3a,b
Inner-product spaces
- Inner-product spaces
- Cauchy-Schwarz inequality
- Parallelogram law
- Pythagoras theorem and its generalization
- The closest point theorem
- Orthogonal projection
- Fourier coefficients
- Hilbert spaces
- Bessels's inequality
- Orthogonal bases and Parseval's identity
- Gram-Schmidt orthogonalization
Relevant exam problems: 2004-5, 2006-4, 2013-5a
Linear transformations: finite dimensional spaces
- Linear transformations and matrices
- Compositions of linear transformations
- Change-of-basis matrix and similar matrices
- Rank of a linear transformation
- Nullity-rank theorem
- Eigenvalues, eigenspaces and generalized eigenspaces
- Caley-Hamilton theorem
Relevant exam problems: 2011-3,
Matrix decompositions
- QR-decomposition
- Jordan normal form and its application to linear systems of ODE
- SVD-decomposition and pseudoinverse
Old exam problems for this section: 2003-3, 2004-3, 2005-2, 2006-2
Linear transformations: vector and normed spaces
- The vector space \(L(X,Y)\)
- Kernels and ranges of linear transformations
- Bounded linear transformations between normed spaces
- Norm of a linear transformation
- The space \(B(X,Y)\)
- Limits of operators
- Power series of operators
- \((I-T)^{-1}\)
- Exponential of an operator
Relevant exam problems: 2005-3c, 2013K-5,
Linear transformations: Hilbert spaces
- Riesz representation theorem
- Adjoint operators
- Self-adjoint and normal operators
- Unitary operators
- Spectral theorem for self-adjoint operators (in finite dimensions)
- Positive definite operators
Relevant exam problems: 2003-4, 2007-5, 2008-5a,b, 2010-5, 2012-5