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About Some Monge–Kantovorich Type Norm and Their Applications to the Theory of Fractals

Author

Listed:
  • Ion Mierluș-Mazilu

    (Department of Mathematics and Computer Science, Technical University of Civil Engineering of Bucharest, 020396 Bucharest, Romania)

  • Lucian Niță

    (Department of Mathematics and Computer Science, Technical University of Civil Engineering of Bucharest, 020396 Bucharest, Romania)

Abstract

If X is a Hilbert space, one can consider the space cabv ( X ) of X valued measures defined on the Borel sets of a compact metric space, having a bounded variation. On this vector measures space was already introduced a Monge–Kantorovich type norm. Our first goal was to introduce a Monge–Kantorovich type norm on cabv ( X ) , where X is a Banach space, but not necessarily a Hilbert space. Thus, we introduced here the Monge–Kantorovich type norm on cabv L q ( [ 0 , 1 ] ) , ( 1 < q < ∞ ) . We obtained some properties of this norm and provided some examples. Then, we used the Monge–Kantorovich norm on cabv K n ( K being R or C ) to obtain convergence properties for sequences of fractal sets and fractal vector measures associated to a sequence of iterated function systems.

Suggested Citation

  • Ion Mierluș-Mazilu & Lucian Niță, 2022. "About Some Monge–Kantovorich Type Norm and Their Applications to the Theory of Fractals," Mathematics, MDPI, vol. 10(24), pages 1-14, December.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:24:p:4825-:d:1007719
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