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On the Construction of the Field of Reals by Means of Functional Equations and Their Stability and Related Topics

In: Developments in Functional Equations and Related Topics

Author

Listed:
  • Jens Schwaiger

    (University of Graz)

Abstract

There are certain approaches to the construction of the field of real numbers which do not refer to the field of rationals. Two of these ideas are closely related to stability investigations for the Cauchy equation and for some homogeneity equation. The a priory different subgroups of β„€ β„€ $$\mathbb{Z}^{\mathbb{Z}}$$ used are shown to be more or less identical. Extension of these investigations shows that given a commutative semigroup G and a normed space X with completion X c the group Hom(G, X c ) is isomorphic to π’œ ( G , X ) βˆ• ℬ ( G , X ) $$\mathcal{A}(G,X)/\mathcal{B}(G,X)$$ where ℬ ( G , X ) $$\mathcal{B}(G,X)$$ is the subgroup of X G of all bounded functions and π’œ ( G , X ) $$\mathcal{A}(G,X)$$ the subgroup of those f: G β†’ X for which the Cauchy difference (x, y) ↦ f(x + y) βˆ’ f(x) βˆ’ f(y) is bounded. The space Hom ( β„• , X c ) $$\text{Hom}(\mathbb{N},X_{c})$$ may be identified with X c itself. With this in mind, we are able to show directly that π’œ ( β„• , X ) βˆ• ℬ ( β„• , X ) $$\mathcal{A}(\mathbb{N},X)/\mathcal{B}(\mathbb{N},X)$$ is a completion of the normed space X.

Suggested Citation

  • Jens Schwaiger, 2017. "On the Construction of the Field of Reals by Means of Functional Equations and Their Stability and Related Topics," Springer Optimization and Its Applications, in: Janusz BrzdΔ™k & Krzysztof CiepliΕ„ski & Themistocles M. Rassias (ed.), Developments in Functional Equations and Related Topics, chapter 0, pages 275-295, Springer.
    • Handle: RePEc:spr:spochp:978-3-319-61732-9_12
    DOI: 10.1007/978-3-319-61732-9_12
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