Banach Journal of Mathematical Analysis

On certain uniformly open multilinear mappings

Marek Balcerzak, Ehrhard Behrends, and Filip Strobin

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We obtain two results stating the uniform openness of bilinear operators and multilinear functionals. The first result deals with Banach spaces Lp:=LKp (over K{R,C}) and pointwise multiplication from Lp×Lq to Lr (where 1/p+1/q=1/r). The second result is concerned with the nontrivial n-linear functionals from the product X1××Xn of normed spaces (over K{R,C}) to the field K.

Article information

Source
Banach J. Math. Anal., Volume 10, Number 3 (2016), 482-494.

Dates
Received: 13 July 2015
Accepted: 2 October 2015
First available in Project Euclid: 13 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1463153912

Digital Object Identifier
doi:10.1215/17358787-3599741

Mathematical Reviews number (MathSciNet)
MR3504181

Zentralblatt MATH identifier
1356.46014

Subjects
Primary: 46B25: Classical Banach spaces in the general theory
Secondary: 47A06: Linear relations (multivalued linear operators) 47A07: Forms (bilinear, sesquilinear, multilinear) 54C10: Special maps on topological spaces (open, closed, perfect, etc.)

Keywords
multiplication uniformly open mapping $L_{p}$ spaces multilinear functionals

Citation

Balcerzak, Marek; Behrends, Ehrhard; Strobin, Filip. On certain uniformly open multilinear mappings. Banach J. Math. Anal. 10 (2016), no. 3, 482--494. doi:10.1215/17358787-3599741. https://projecteuclid.org/euclid.bjma/1463153912


Export citation

References

  • [1] M. Balcerzak, A. Majchrzycki, and A. Wachowicz, Openness of multiplication in some function spaces, Taiwanese J. Math. 17 (2013), no. 3, 1115–1126.
  • [2] M. Balcerzak, A. Wachowicz, and W. Wilczyński, Multiplying balls in $C[0,1]$, Studia Math. 170 (2005), no. 2, 203–209.
  • [3] E. Behrends, Products of $n$ open subsets in the space of continuous functions on $[0,1]$, Studia Math. 204 (2011), no. 1, 73–95.
  • [4] P. J. Cohen, A counterexample to the closed graph theorem for bilinear maps, J. Funct. Anal. 16 (1974), 235–239.
  • [5] L. Holá, A. K. Mirmostafaee, and Z. Piotrowski, Points of openness and closedness of some mappings, Banach J. Math. Anal. 9 (2015), no. 1, 243–252.
  • [6] C. Horowitz, An elementary counterexample to the open mapping principle for bilinear maps, Proc. Amer. Math. Soc. 53 (1975), no. 2, 293–294.
  • [7] A. Komisarski, A connection between multiplication in $C(X)$ and the dimension of $X$, Fund. Math. 189 (2006), no. 2, 149–154.
  • [8] K. Kuratowski, Topology, I, Academic Press, New York 1966.
  • [9] R. Li, S. Zhong, and C. Swartz, An open mapping theorem without continuity and linearity, Topology Appl. 157 (2010), no. 13, 2086–2093.
  • [10] A. Majchrzycki, Openness of bilinear mappings and some properties of integrable functions (in Polish), Ph.D. dissertation, Łódź University of Technology, Łódź, 2015.
  • [11] W. Rudin, Function Theory in Polydiscs, Benjamin, New York, 1969.
  • [12] W. Rudin, Functional Analysis, 2nd ed., Internat. Ser. Pure Appl. Math., McGraw-Hill, New York, 1991.