Banach Journal of Mathematical Analysis

On certain uniformly open multilinear mappings

Marek Balcerzak, Ehrhard Behrends, and Filip Strobin

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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

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

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

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Zentralblatt MATH identifier

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.)

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


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.

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  • [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.