Total set

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

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

See also

  • Kadec norm – All infinite-dimensional, separable Banach spaces are homeomorphicPages displaying short descriptions of redirect targets
  • Degenerate bilinear form – Possible x & y for x-E conjugatesPages displaying wikidata descriptions as a fallback
  • Dual system
  • Topologies on spaces of linear maps

References

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