Total set

In functional analysis, a total set (also called a complete set) in a vector space is a set of linear functionals $T$ with the property that if a vector $x\in X$ satisfies $f(x)=0$ for all $f\in T,$ then $x=0$ is the zero vector.^[1]

In a more general setting, a subset $T$ of a topological vector space $X$ is a total set or fundamental set if the linear span of $T$ is dense in $X.$ ^[2]

References

↑ Klauder, John R. (2010). A Modern Approach to Functional Integration. Springer Science & Business Media. p. 91. ISBN 9780817647902.
↑ Lomonosov, L. I. "Total set". Encyclopedia of Mathematics. Springer. Retrieved 14 September 2014.

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[Klauder2010-1] Klauder, John R. (2010). A Modern Approach to Functional Integration. Springer Science & Business Media. p. 91. ISBN 9780817647902.

[Lomonosov2011-2] Lomonosov, L. I. "Total set". Encyclopedia of Mathematics. Springer. Retrieved 14 September 2014.

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v t e Duality and spaces of linear maps
Basic concepts	Dual space Dual system Dual topology Duality Operator topologies Polar set Polar topology Topologies on spaces of linear maps
Topologies	Norm topology Dual norm Ultraweak/Weak-* Weak polar operator in Hilbert spaces Mackey Strong dual polar topology operator Ultrastrong
Main results	Banach–Alaoglu Mackey–Arens
Maps	Transpose of a linear map
Subsets	Saturated family Total set
Other concepts	Biorthogonal system

v t e Topological vector spaces (TVSs)
Basic concepts	Banach space Completeness Continuous linear operator Linear functional Fréchet space Linear map Locally convex space Metrizability Operator topologies Topological vector space Vector space
Main results	Anderson–Kadec Banach–Alaoglu Closed graph theorem F. Riesz's Hahn–Banach (hyperplane separation Vector-valued Hahn–Banach) Open mapping (Banach–Schauder) Bounded inverse Uniform boundedness (Banach–Steinhaus)
Maps	Bilinear operator form Linear map Almost open Bounded Continuous Closed Compact Densely defined Discontinuous Topological homomorphism Functional Linear Bilinear Sesquilinear Norm Seminorm Sublinear function Transpose
Types of sets	Absolutely convex/disk Absorbing/Radial Affine Balanced/Circled Banach disks Bounding points Bounded Complemented subspace Convex Convex cone (subset) Linear cone (subset) Extreme point Pre-compact/Totally bounded Prevalent/Shy Radial Radially convex/Star-shaped Symmetric
Set operations	Affine hull (Relative) Algebraic interior (core) Convex hull Linear span Minkowski addition Polar (Quasi) Relative interior
Types of TVSs	Asplund B-complete/Ptak Banach (Countably) Barrelled BK-space (Ultra-) Bornological Brauner Complete Convenient (DF)-space Distinguished F-space FK-AK space FK-space Fréchet tame Fréchet Grothendieck Hilbert Infrabarreled Interpolation space K-space LB-space LF-space Locally convex space Mackey (Pseudo)Metrizable Montel Quasibarrelled Quasi-complete Quasinormed (Polynomially Semi-) Reflexive Riesz Schwartz Semi-complete Smith Stereotype (B Strictly Uniformly) convex (Quasi-) Ultrabarrelled Uniformly smooth Webbed With the approximation property
Category

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Total set

See also

References