Hilbert Space as Infinite-Dimensional Inner Product Space
HilbertSpace InnerProduct InfiniteDimension
03:30 Abide by Reason introduces Hilbert spaces for learners who know Euclidean vectors and dot products but are unsure how these ideas extend to abstract, infinite-dimensional settings.
Completeness, Cauchy Sequences, and Completion Analogy for Hilbert Spaces
Completeness CauchySequence SpaceCompletion
09:00 This note is for learners grappling with the abstract completeness requirement in the definition of Hilbert spaces.
Examples of Hilbert Spaces: L² Sequences and L² Functions
L2Space SequenceSpace FunctionSpaces
15:00 Abide by Reason provides concrete Hilbert space examples for viewers who want to see beyond abstract definitions.
Orthonormal Bases in Hilbert Spaces and Fourier Expansions
OrthonormalBasis FourierSeries FunctionCoordinates
19:00 This note targets learners connecting Hilbert space theory to familiar Fourier series and the idea of representing functions by “coordinates.”
Hilbert Spaces in Quantum Mechanics and Wave Functions
QuantumHilbertSpace WaveFunction L2R
22:00 Abide by Reason connects the abstract Hilbert space concept to quantum mechanics for viewers curious why physicists care about (L^2) spaces.