The intersection convolution of relations and the Hahn-Banach type theorems

Árpád Száz

Annales Polonici Mathematici (1998)

  • Volume: 69, Issue: 3, page 235-249
  • ISSN: 0066-2216

Abstract

top
By introducing the intersection convolution of relations, we prove a natural generalization of an extension theorem of B. Rodrí guez-Salinas and L. Bou on linear selections which is already a substantial generalization of the classical Hahn-Banach theorems. In particular, we give a simple neccesary and sufficient condition in terms of the intersection convolution of a homogeneous relation and its partial linear selections in order that every partial linear selection of this relation can have an extension to a total linear selection.

How to cite

top

Árpád Száz. "The intersection convolution of relations and the Hahn-Banach type theorems." Annales Polonici Mathematici 69.3 (1998): 235-249. <http://eudml.org/doc/270701>.

@article{ÁrpádSzáz1998,
abstract = {By introducing the intersection convolution of relations, we prove a natural generalization of an extension theorem of B. Rodrí guez-Salinas and L. Bou on linear selections which is already a substantial generalization of the classical Hahn-Banach theorems. In particular, we give a simple neccesary and sufficient condition in terms of the intersection convolution of a homogeneous relation and its partial linear selections in order that every partial linear selection of this relation can have an extension to a total linear selection.},
author = {Árpád Száz},
journal = {Annales Polonici Mathematici},
keywords = {intersection convolution; additive and homogeneous relations; linear selections; binary intersection property; Hahn-Banach theorems; Hahn-Banach extension theorem; injectivity of Banach spaces; Hausdorff's maximality principle; Nachbin's Theorem},
language = {eng},
number = {3},
pages = {235-249},
title = {The intersection convolution of relations and the Hahn-Banach type theorems},
url = {http://eudml.org/doc/270701},
volume = {69},
year = {1998},
}

TY - JOUR
AU - Árpád Száz
TI - The intersection convolution of relations and the Hahn-Banach type theorems
JO - Annales Polonici Mathematici
PY - 1998
VL - 69
IS - 3
SP - 235
EP - 249
AB - By introducing the intersection convolution of relations, we prove a natural generalization of an extension theorem of B. Rodrí guez-Salinas and L. Bou on linear selections which is already a substantial generalization of the classical Hahn-Banach theorems. In particular, we give a simple neccesary and sufficient condition in terms of the intersection convolution of a homogeneous relation and its partial linear selections in order that every partial linear selection of this relation can have an extension to a total linear selection.
LA - eng
KW - intersection convolution; additive and homogeneous relations; linear selections; binary intersection property; Hahn-Banach theorems; Hahn-Banach extension theorem; injectivity of Banach spaces; Hausdorff's maximality principle; Nachbin's Theorem
UR - http://eudml.org/doc/270701
ER -

References

top
  1. [1] J. Abreu and A. Etchebery, Hahn-Banach and Banach-Steinhaus theorems for convex processes, Period. Math. Hungar. 20 (1989), 289-297. Zbl0664.46012
  2. [2] R. Arens, Operational calculus of linear relations, Pacific J. Math. 11 (1961), 9-23. Zbl0102.10201
  3. [3] G. Buskes, The Hahn-Banach Theorem surveyed, Dissertationes Math. 327 (1993). Zbl0808.46003
  4. [4] J. Dieudonné, History of Functional Analysis, North-Holland Math. Stud. 49, North-Holland, Amsterdam, 1981. Zbl0478.46001
  5. [5] B. Fuchssteiner und J. Horváth, Die Bedeutung der Schnitteigenschaften beim Hahn-Banachschen Satz, Jahrbuch Überblicke Math. (BI, Mannheim) 1979, 107-121. Zbl0405.46001
  6. [6] B. Fuchssteiner and W. Lusky, Convex Cones, North-Holland Math. Stud. 56, North-Holland, Amsterdam, 1981. Zbl0478.46002
  7. [7] Z. Gajda, A. Smajdor and W. Smajdor, A theorem of the Hahn-Banach type and its applications, Ann. Polon. Math. 57 (1992), 243-252. Zbl0774.46003
  8. [8] G. Godini, Set-valued Cauchy functional equation, Rev. Roumaine Math. Pures Appl. 20 (1975), 1113-1121. Zbl0322.39013
  9. [9] D. B. Goodner, Projections in normed linear spaces, Trans. Amer. Math. Soc. 69 (1950), 89-108. Zbl0041.23203
  10. [10] M. Hasumi, The extension property of complex Banach spaces, Tôhoku Math. J. 10 (1958), 135-142. Zbl0087.10901
  11. [11] L. Holá and P. Maličký, Continuous linear selectors of linear relations, Acta Math. Univ. Comenian. 48-49 (1986), 153-157. Zbl0629.46005
  12. [12] J. Horváth, Some selected results of professor Baltasar Rodrí guez-Salinas, Rev. Mat. Univ. Complut. Madrid 9 (1996), 23-72. Zbl0867.01040
  13. [13] O. Hustad, A note on complex 𝓟₁ spaces, Israel J. Math. 16 (1973), 117-119. Zbl0284.46012
  14. [14] A. W. Ingleton, The Hahn-Banach theorem for non-Archimedean-valued fields, Proc. Cambridge Philos. Soc. 48 (1952), 41-45. Zbl0046.12001
  15. [15] A. D. Ioffe, A new proof of the equivalence of the Hahn-Banach extension and the least upper bound properties, Proc. Amer. Math. Soc. 82 (1981), 385-389. Zbl0469.46005
  16. [16] J. L. Kelley, Banach spaces with the intersection property, Trans. Amer. Math. Soc. 72 (1952), 323-326. Zbl0046.12002
  17. [17] J. L. Kelley, General Topology, Van Nostrand Reinhold, New York, 1955. 
  18. [18] S. Mac Lane, Homology, Springer, Berlin, 1963. 
  19. [19] L. Nachbin, A theorem of the Hahn-Banach type for linear transformations, Trans. Amer. Math. Soc. 68 (1950), 28-46. Zbl0035.35402
  20. [20] K. Nikodem, Additive selections of additive set-valued functions, Zb. Rad. Prirod.-Mat. Fak. Univ. u Novom Sadu Ser. Mat. 18 (1988), 143-148. Zbl0686.46008
  21. [21] Zs. Páles, Linear selections for set-valued functions and extension of bilinear forms, Arch. Math. (Basel) 62 (1994), 427-432. Zbl0797.46005
  22. [22] B. Rodríguez-Salinas and L. Bou, A Hahn-Banach theorem for arbitrary vector spaces, Boll. Un. Mat. Ital. 10 (1974), 390-393. Zbl0309.47006
  23. [23] W. Smajdor, Subadditive and subquadratic set-valued functions, Prace Nauk. Uniw. Śląsk. Katowic. 889 (1987), 73 pp. Zbl0626.54019
  24. [24] W. Smajdor and J. Szczawińska, A theorem of the Hahn-Banach type, Demonstratio Math. 28 (1995), 155-160. Zbl0832.46003
  25. [25] T. Strömberg, The operation of infimal convolution, Dissertationes Math. 352 (1996). Zbl0858.49010
  26. [26] Á. Száz, Pointwise limits of nets of multilinear maps, Acta Sci. Math. (Szeged) 55 (1991), 103-117. Zbl0783.46004
  27. [27] Á. Száz, Foundations of Linear Analysis, Inst. Mat. Inf., Lajos Kossuth University Debrecen 1996, 200 pp. (Unfinished lecture notes in Hungarian). 
  28. [28] Á. Száz and G. Száz, Additive relations, Publ. Math. Debrecen 20 (1973), 259-272. Zbl0362.08002
  29. [29] Á. Száz and G. Száz, Linear relations, Publ. Math. Debrecen 27 (1980), 219-227. Zbl0467.46015
  30. [30] J. Zowe, Sandwich theorems for convex operators with values in an ordered vector space, J. Math. Anal. Appl. 66 (1978), 282-396. Zbl0389.46003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.