User:Mgkrupa/Functional analysis proposal

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Functional analysis (topics)

Linear algebra

Basic concepts	Vector space Linear operators Functionals
Types of sets	Absolutely convex Absorbing Affine Balanced/Circled Convex Convex cone (subset) Linear cone (subset) Radial Radially convex/Star-shaped Symmetric
Subsets / set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Maps	Bilinear (form operator) and Sesquilinear forms Functionals Operator Transpose

Duality theory and Spaces of Linear and Bilinear Maps

Basic concepts	Dual Dual space Topology of uniform convergence
Topologies	Dual norm Ultraweak/Weak-* Weak (polar operator) Mackey Strong (polar operator) Ultrastrong
Main results	Banach–Alaoglu Mackey–Arens

Hilbert space theory

Basic concepts	Adjoint Inner product and Semi-inner-product Hilbert space and Prehilbert space
Main results	Bessel's inequality Cauchy–Schwarz inequality Riesz representation
Other results	Parseval's identity Polarization identity
Operators	Compact operator on Hilbert space Densely defined Hilbert–Schmidt Normal Self-adjoint Unitary
Examples	Segal–Bargmann F

Spectral theory and C^* algebras

Basic concepts	Banach algebra C-algebra Spectrum Spectrum of a C-algebra Spectral radius Operator space
Main results	Gelfand–Mazur theorem Gelfand–Naimark theorem Polar decomposition Singular value decomposition Spectral theorem
Applications	Spectral theory of ordinary differential equations

Normed and Banach space theory

Basic concepts

Dual

Main results

Other results

Operators

Examples (list)

Sequence spaces	Bs space c space Banach coordinate BK Hilbert H ℓ^p (ℓ^{${\infty }$})
L^p spaces and generalizations	Besov $B_{p,q}^{s}(\mathbf {R} )$ Birnbaum–Orlicz Bounded mean oscillation BMO Hardy H^p L^p ( $L^{\infty }$ weighted) Morrey–Campanato $L^{\lambda ,p}(\Omega )$ Sobolev W^k,p (Sobolev inequality) Triebel–Lizorkin Wiener amalgam $W(X,L^{p})$
Spaces of measures	$ba(\Sigma )$
Other spaces	Auxiliary normed spaces

Bornology and boundedness

Basic concepts	Barrelled space Bounded set Bornological space
Operators	(Un)Bounded operator

Vector lattices and Riesz space theory

Basic concepts	Banach lattice Riesz space vector lattice
Operators	positive
Main results	Freudenthal spectral

Topological tensor products and Nuclear spaces

Basic concepts	Auxiliary normed spaces Nuclear space Tensor product (of Hilbert spaces)
Topologies	Inductive tensor product Injective tensor product Projective tensor product
Operators	Integral Nuclear (between Banach spaces) Trace class

Analysis in Topological vector spaces

Basic concepts	Abstract Wiener space Bochner space Convex series
Derivatives	Differentiation in Fréchet spaces Fréchet Gateaux functional holomorphic quasi
Measurability	Measures (Lebesgue Projection-valued Vector) Weakly / Strongly measurable function
Integrals	Bochner Dunford Gelfand–Pettis regulated Paley–Wiener weak
Main results	Inverse function theorem (Nash–Moser theorem)

Applications
Differential operator Finite element method Mathematical formulation of quantum mechanics Ordinary Differential Equations (ODEs) Validated numerics

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