In mathematics more specifically in functional analysis, a subset T of a topological vector space X is said to be a total subset of X if the linear span of T is a dense subset of X.[1] This condition arises frequently in many theorems of functional analysis.
Examples
Unbounded self-adjoint operators on Hilbert spaces are defined on total subsets.
See also
Dense subset
Positive linear operator
Topological vector spaces
References
Schaefer 1999, p. 80.
Schaefer, Helmut H. (1999). Topological Vector Spaces. GTM. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
Functional analysis (topics – glossary)
Spaces
Hilbert space Banach space Fréchet space topological vector space
Theorems
Hahn–Banach theorem closed graph theorem uniform boundedness principle Kakutani fixed-point theorem Krein–Milman theorem min-max theorem Gelfand–Naimark theorem Banach–Alaoglu theorem
Operators
bounded operator compact operator adjoint operator unitary operator Hilbert–Schmidt operator trace class unbounded operator
Algebras
Banach algebra C*-algebra spectrum of a C*-algebra operator algebra group algebra of a locally compact group von Neumann algebra
Open problems
invariant subspace problem Mahler's conjecture
Applications
Besov space Hardy space spectral theory of ordinary differential equations heat kernel index theorem calculus of variation functional calculus integral operator Jones polynomial topological quantum field theory noncommutative geometry Riemann hypothesis
Advanced topics
locally convex space approximation property balanced set Schwartz space weak topology barrelled space Banach–Mazur distance Tomita–Takesaki theory
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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