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Functional Analysis
Explore norms and completeness; duality and convergence; operator theory. Concepts and connections are available independently of video coverage.
Prerequisites and connections
Vector spaces
A vector space specifies addition and scalar multiplication satisfying the vector-space axioms over a chosen field.
2 reviewed resources
Norms and completeness
Normed spaces
A norm measures vector size with homogeneity, separation and the triangle inequality.
Video explanations to be addedBanach spaces
A Banach space is a normed vector space complete in its induced metric.
Video explanations to be addedHilbert spaces
A Hilbert space is complete under a norm induced by an inner product.
Video explanations to be addedBounded linear operators
A linear operator bounded by a constant times input norm is continuous.
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Duality and convergence
Dual spaces
Continuous linear functionals on a normed space form its continuous dual.
Video explanations to be addedHahn–Banach theorem
Appropriately bounded linear functionals extend from subspaces while preserving their bounds.
Video explanations to be addedWeak convergence
A sequence converges weakly when every continuous linear functional converges on it.
Video explanations to be addedRiesz representation in Hilbert spaces
Each continuous linear functional on a Hilbert space is represented by an inner product with a unique vector.
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Operator theory
Compact operators
A compact operator maps bounded sets into relatively compact sets.
Video explanations to be addedOperator spectrum
The spectrum consists of scalars where subtracting the scalar identity fails to have a bounded inverse.
Video explanations to be addedSelf-adjoint operators
For bounded Hilbert-space operators, self-adjointness means equality with the adjoint.
Video explanations to be addedSpectral theorem
Suitable self-adjoint or normal operators admit spectral decompositions; the precise form depends on the setting.
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