Mathematics / research
Operators, spectra, and infinite dimensions
Extend linear algebra carefully to Hilbert spaces, compact and unbounded operators, resolvents, semigroups, and variational spectra.
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Module 01 research
Operators, spectra, and infinite dimensions
Extend linear algebra carefully to Hilbert spaces, compact and unbounded operators, resolvents, semigroups, and variational spectra.
- 01
Hilbert spaces and orthogonal projection
Find the nearest admissible vector and see why closedness matters in infinite dimensions.
28 min - 02
Bounded operators and adjoints
Control linear amplification and define the adjoint without relying on a matrix transpose.
28 min - 03
Compact operators and the Fredholm alternative
Understand finite-rank approximation, exceptional eigenvalues, and compatibility conditions for solvability.
31 min - 04
Spectra and resolvents
Replace a determinant test with invertibility, stability, and continuous-spectrum examples.
30 min - 05
The spectral theorem for self-adjoint operators
Understand spectral projections, functional calculus, and the limits of an eigenvector-only picture.
33 min - 06
Unbounded operators and their domains
Treat the domain as part of the operator and distinguish symmetric from self-adjoint realizations.
33 min - 07
Sturm–Liouville problems
Connect weighted inner products and boundary forms to real eigenvalues and normal modes.
32 min - 08
Operator semigroups and evolution
Connect generators, domains, and strongly continuous solution operators for time-dependent problems.
32 min - 09
Variational eigenvalue principles
Estimate low eigenvalues through trial spaces while tracking the assumptions behind upper bounds.
32 min - 10
Distributions and rigged Hilbert spaces
Give generalized states a test-function meaning without pretending they are normalizable vectors.
33 min