URL study guide
https://tue.osiris-student.nl/onderwijscatalogus/extern/cursus?cursuscode=2WAF0&collegejaar=2025&taal=enDescription
Metric spaces: basis notions of topology and convergence,completeness of metric spaces, compactness.
Normed vector spaces and Banach spaces: linear operators
and linear functionals, bounded linear operators and continuity,
range, null space and inverse operator, dual space.
Inner product spaces and Hilbert spaces: inner products and
orthogonality, (total) orthonormal sequences, Fourier series,
Bessel inequality and Parseval relation, Riesz representation
theorem, Hilbert adjoint operator.
Approximation of periodic functions with Fourier series:
pointwise and uniform convergence of Fourier series.
Objectives
- a profound knowledge of the topological structure of metricspaces, and being able to provide proofs of elementary properties
of metric spaces.
- being able to carry out proofs for well-known metric spaces,
like l_p and L_p spaces, or the space of continuous functions
on a compact interval.
- being able to prove whether a metric space is complete,
and whether a subset of a metric space is compact.
- a profound knowledge of both the algebraic and topological
structure of normed vector spaces and Banach spaces, and being
able to provide proofs of elementary properties of these spaces.
- being able to check whether an operator between normed vector
spaces is linear and/or continuous/bounded. Being able to compute
the norm of a bounded linear operator. Being able to determine
the range and null space of a linear operator, and if applicable
its inverse.
- understand the notion of dual spaces, and being able to
determine the dual space of a Banach space. Know that the dual
space of a normed vector space is complete.
- a profound knowledge of the algebraic, topological and geometric
structure of inner product spaces and Hilbert spaces, and being
able to provide proofs of elementary properties of these spaces.
- understand the relationship between orthogonality and minimal
distance. Being able to carry out proofs concerning orthogonal
complements and orthogonal projections.
- know the notion of (total) orthonormal sets and sequences,
including the relationship with Fourier series, the
Bessel inequality and Parseval relation. Being able to use
Fourier series for approximation in Hilbert spaces.
- being able to determine the classical Fourier series of
simple periodic functions, and to check their convergence
properties, i.e. being able to investigate the pointwise or
uniform convergence of classical Fourier series.
- knowledge of the Riesz representation theorem for linear
functionals on Hilbert spaces, and being able to apply
this theorem, for example in the definition of the
Hilbert adjoint operator