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2 results

  • 20 - Some Applications of the Hahn–Banach Theorem

  • 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 219-227

  • Summary

    We give some important applications of the Hahn-Banach Theorem. We prove the existence of a support functional and hence that X^* separates points in X. Then we prove the existence of a functional that encodes the distance from a linear subspace, which is an important ingredient in a number of subsequent proofs. We show that separability of X^* implies separability of X, define the Banach adjoint of a linear map (between Banach spaces), and prove the existence of ‘generalised Banach limits’.

  • Higher Connectedness Properties of Support Points and Functionals of Convex Sets

  • Carlo Alberto De Bernardi
  • Journal:
    Canadian Journal of Mathematics / Volume 65 / Issue 6 / 01 December 2013
    Published online by Cambridge University Press:
    20 November 2018, pp. 1236-1254
    Print publication:
    01 December 2013
    • Article
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  • We prove that the set of all support points of a nonempty closed convex bounded set $C$ in a real infinite-dimensional Banach space $X$ is $\text{AR}$($\sigma $-compact) and contractible. Under suitable conditions, similar results are proved also for the set of all support functionals of $C$ and for the domain, the graph, and the range of the subdifferential map of a proper convex lower semicontinuous function on $X$.