Section
Functional Analysis
Locally convex spaces, distributions, direct integrals, and bounded and unbounded operators.
Core idea
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Functional analysis
Banach, Hilbert, and bounded-operator foundations
- Adjoint of a bounded operator
- Banach algebra
- Banach–Alaoglu theorem
- Bounded commutator
- Bounded linear operator between normed spaces
- Closed graph theorem
- Commutator of possibly unbounded operators
- Finite-rank operator
- Fredholm operator
- Open mapping theorem
- Partial isometry
- Reflexive Banach space
- Riesz representation theorem for Hilbert spaces
- Hilbert–Schmidt operator
- Schatten-class operator
- Singular values of a compact operator
- Uniform boundedness principle
- Unitary operator
- Z/2-graded Hilbert space
Unbounded operators
- Adjoint of a densely defined operator
- Borel functional calculus for self-adjoint and normal operators
- Closable operator
- Closed linear operator
- Closed quadratic form
- Closure of a closable operator
- Core of a closed operator
- Deficiency indices
- Densely defined operator
- Essentially self-adjoint operator
- Friedrichs extension
- Graph norm
- Graph of a linear operator
- Operator with compact resolvent
- Point, continuous, and residual spectrum
- Projection-valued measure
- Resolvent identity
- Resolvent set of a closed operator
- Self-adjoint extension
- Self-adjoint unbounded operator
- Spectral integral
- Spectral projection of a self-adjoint or normal operator
- Spectral theorem for unbounded self-adjoint operators
- Spectrum of a closed operator
- Symmetric operator
Spectra and functional calculus
- Holomorphic functional calculus in a Banach algebra
- Invertible element in a Banach algebra
- Resolvent of an element in a Banach algebra
- Spectral radius formula for a Banach algebra
- Spectral radius in a Banach algebra
- Spectrum of an element in a Banach algebra
Locally convex spaces
- Barreled space
- Bornological space
- Bounded subset of a topological vector space
- Continuous linear map between topological vector spaces
- Equicontinuous family of continuous linear maps
- Fréchet space
- Inductive limit of locally convex spaces
- LF-space
- Locally convex space
- Montel space
- Nuclear space
- Projective limit of locally convex spaces
- Strong dual of a locally convex space
- Topological dual
- Topological vector space
- Topology generated by a family of seminorms
- Weak topology on a topological vector space
- Weak-star topology
Schwartz spaces and distributions
- Compactly supported distribution
- Convolution of distributions
- Distribution
- Distributional derivative
- Fourier inversion on Schwartz space
- Fourier transform of tempered distributions
- Fourier transform on Schwartz space
- Order of a distribution
- Regular tempered distribution
- Rigged Hilbert space
- Schwartz kernel theorem
- Schwartz space on Euclidean space
- Singular support of a distribution
- Sobolev space
- Support of a distribution
- Tempered distribution
- Test-function space
- Wavefront set
- Weak derivative
Direct integrals
- Decomposable operator
- Direct integral of Hilbert spaces
- Direct integral of operators
- Measurable field of Hilbert spaces
Integration and function-space additions
- Absolutely convergent series in a Banach space
- Bounded bilinear map between normed spaces
- Continuous Banach-valued functions on a compact space
- Integral operator with a measurable kernel
Evolution-equation prerequisites
Analytic dependence and coefficient estimates
- Product estimate in a two-index analytic coefficient space
- Radial operator bounds in an analytic coefficient norm
- Holomorphic realization of weighted analytic coefficients
- Operator inversion from summable bounds on powers
- A mixed-derivative estimate after radial inversion
- Two-index analytic coefficient space
Estimates with derivative loss
Cones, averages and moment constraints
Sobolev spaces and pressure analysis
- Dirac delta distribution
- Fourier Sobolev space on Euclidean space
- Integrable functions in negative Sobolev spaces
- Local Sobolev space
- Multiplication operator
- Smooth compact-support approximation in Euclidean Sobolev space
- Derivative bound between Fourier Sobolev spaces
- Sobolev embedding into bounded continuous derivatives
- Three-dimensional Sobolev inequality
- Sobolev estimate with a scaled cutoff