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Linear Continuous Representations

In: Continuity Theory

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  • Louis Nel

    (Carleton University, School of Mathematics and Statistics)

Abstract

This chapter deals with a variety of linear continuous representations. It begins with representation of the gauged reflection of a paradual C X = C [ X , 𝕂 ] $$\mathtt{C}\,X = \mathsf{C}[X, \mathbb{K}]$$ . This gives valuable insight into the nature of CV-functionals on C X. Further valuable insight comes from a representation of CV-functionals on C Q with compact Q via free CV-functionals. These results pave the way for a (new) proof that every C X is reflexive. This again leads to the noteworthy result that all cGV-spaces (complete locally convex topological vector spaces) are reflexive, whence cGV is dually equivalent to a category in which all spaces are complete and locally compact. Then, elaborating on the preliminary representation via free functionals, we derive a Riesz-Radon representation of Δ C X $$\Delta \mathtt{C}\,X$$ for all C-spaces X, thus generalizing the earlier representation obtained for compact X.

Suggested Citation

  • Louis Nel, 2016. "Linear Continuous Representations," Springer Books, in: Continuity Theory, chapter 0, pages 403-415, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-31159-3_13
    DOI: 10.1007/978-3-319-31159-3_13
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