Rad HAZU, Matematičke znanosti, Vol. 22 (2018), 145-170.

THE QUOTIENT SHAPES OF lp AND Lp SPACES

Nikica Uglešić

Veli Råt, Dugi Otok, Hrvatska
e-mail: nuglesic@unizd.hr


Abstract.   All lp spaces (over the same field), p ≠ ∞, have the finite quotient shape type of the Hilbert space l2. It is also the finite quotient shape type of all the subspaces lp(p'), p < p' ≤ ∞, as well as of all their direct sum subspaces F0N(p'), 1 ≤ p' ≤ ∞. Furthermore, their countable and finite quotient shape types coincide. Similarly, for a given positive integer, all Lp spaces (over the same field) have the finite quotient shape type of the Hilbert space L2, and their countable and finite quotient shape types coincide. Quite analogous facts hold true for the (special type of) Sobolev spaces (of all appropriate real functions).

2010 Mathematics Subject Classification.   54E99, 55P55.

Key words and phrases.   Normed (Banach, Hilbert vectorial) space, quotient normed space, lp space, Lp space, Sobolev space, (algebraic) dimension, (infinite) cardinal, (general) continuum hypothesis, quotient shape.


Full text (PDF) (free access)

DOI: http://doi.org/10.21857/9xn31cr62y


References:

  1. K. Borsuk, Concerning homotopy properties of compacta, Fund. Math. 62 (1968), 223–254.
    MathSciNet     CrossRef

  2. K. Borsuk, Theory of Shape, Monografie Matematyczne 59, Polish Scientific Publishers, Warszawa, 1975.
    MathSciNet

  3. J.-M. Cordier and T. Porter, Shape Theory: Categorical Methods of Approximation, Ellis Horwood Ltd., Chichester, 1989. (Dover edition, 2008.)
    MathSciNet

  4. J. Dugundji, Topology, Allyn and Bacon, Inc. Boston, 1978.
    MathSciNet

  5. J. Dydak and J. Segal, Shape theory: An introduction, Lecture Notes in Math. 688, Springer-Verlag, Berlin, 1978.
    MathSciNet

  6. D.A. Edwards and H.M. Hastings, Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Math. 542, Springer-Verlag, Berlin, 1976.
    MathSciNet

  7. H. Herlich and G. E. Strecker, Category Theory: An Introduction, Allyn and Bacon Inc., Boston, 1973.
    MathSciNet

  8. N. Koceić Bilan and N. Uglešić, The coarse shape, Glas. Mat. Ser. III 42(62) (2007), 145–187.
    MathSciNet     CrossRef

  9. E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1989.
    MathSciNet

  10. S. Kurepa, Funkcionalna analiza : elementi teorije operatora, Školska knjiga, Zagreb, 1990.

  11. S. Mardešić and J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
    MathSciNet

  12. W. Rudin, Functional Analysis, Second Edition, McGraw-Hill, Inc., New York, 1991.
    MathSciNet

  13. N. Uglešić, The shapes in a concrete category, Glas. Mat. Ser. III 51(71) (2016), 255–306.
    MathSciNet     CrossRef

  14. N. Uglešić, On the quotient shape of vectorial spaces, Rad Hrvat. Akad. Znan. Umjet. Mat. Znan. 21 (2017), 179–203.
    MathSciNet     CrossRef

  15. N. Uglešić, On the quotient shape of topological spaces, Topology Appl. 239 (2018), 142–151.
    MathSciNet     CrossRef

  16. N. Uglešić and B. Červar, The concept of a weak shape type, International J. of Pure and Applied Math. 39 (2007), 363–428.
    MathSciNet


Rad HAZU Home Page