Smith space

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In functional analysis and related areas of mathematics, Smith space is a complete compactly generated locally convex space having a compact set which absorbs every other compact set (i.e. for some ).

Smith spaces are named after Marianne Ruth Freundlich Smith, who introduced them[1] as duals to Banach spaces in some versions of duality theory for topological vector spaces. All Smith spaces are stereotype and are in the stereotype duality relations with Banach spaces:[2][3]

  • for any Banach space its stereotype dual space[4] is a Smith space,
  • and vice versa, for any Smith space its stereotype dual space is a Banach space.

See also[edit]

Notes[edit]

  1. ^ Smith 1952.
  2. ^ Akbarov 2003, p. 220.
  3. ^ Akbarov 2009, p. 467.
  4. ^ The stereotype dual space to a locally convex space is the space of all linear continuous functionals endowed with the topology of uniform convergence on totally bounded sets in .

References[edit]

  • Smith, M.F. (1952). "The Pontrjagin duality theorem in linear spaces". Annals of Mathematics. 56 (2): 248–253. doi:10.2307/1969798. JSTOR 1969798.
  • Akbarov, S.S. (2003). "Pontryagin duality in the theory of topological vector spaces and in topological algebra". Journal of Mathematical Sciences. 113 (2): 179–349. doi:10.1023/A:1020929201133.
  • Akbarov, S.S. (2009). "Holomorphic functions of exponential type and duality for Stein groups with algebraic connected component of identity". Journal of Mathematical Sciences. 162 (4): 459–586. arXiv:0806.3205. doi:10.1007/s10958-009-9646-1. (Subscription required (help)).
  • Furber, R.W.J. (2017). Categorical Duality in Probability and Quantum Foundations (PhD). Radboud University.