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3 results

  • 21 - Convex Subsets of Banach Spaces

  • from Part IV - Banach Spaces
  • James C. Robinson, University of Warwick
  • Book:
    An Introduction to Functional Analysis
    Published online:
    21 March 2020
    Print publication:
    12 March 2020, pp 228-239

  • Summary

    As a corollary of the Hahn-Banach Theorem, we show that any two convex sets can be separated using a linear functional; a key ingredient is the definition of the Minkowski functional of a convex set. This separation theorem allows us to give a characterisation of convex sets in terms of their supporting hyperplanes that will be useful later. We then define the closed convex hull of a set, introduce the notion of extreme points in a convex set, and prove the Krein-Milman Theorem: a non-empty compact convex subset of a Banach space is the closed convex hull of its extreme points.

  • SHEAF RECURSION AND A SEPARATION THEOREM

  • NATHANAEL LEEDOM ACKERMAN
  • Journal:
    The Journal of Symbolic Logic / Volume 79 / Issue 3 / September 2014
    Published online by Cambridge University Press:
    18 August 2014, pp. 882-907
    Print publication:
    September 2014

  • Define a second order tree to be a map between trees (with fixed codomain). We show that many properties of ordinary trees have analogs for second order trees. In particular, we show that there is a notion of “definition by recursion on a well-founded second order tree” which generalizes “definition by transfinite recursion”. We then use this new notion of definition by recursion to prove an analog of Lusin’s Separation theorem for closure spaces of global sections of a second order tree.

  • ON A SEPARATION THEOREM INVOLVING THE QUASI-RELATIVE INTERIOR

  • F. Cammaroto, B. Di Bella
  • Journal:
    Proceedings of the Edinburgh Mathematical Society / Volume 50 / Issue 3 / October 2007
    Published online by Cambridge University Press:
    08 January 2008, pp. 605-610
    • Article
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  • We establish two separation theorems in which the classic interior is replaced by the quasi-relative interior.