## Banach Journal of Mathematical Analysis

### Spaceability in norm-attaining sets

#### Abstract

We study the existence of infinite-dimensional vector spaces in the sets of norm-attaining operators, multilinear forms, and polynomials. Our main result is that, for every set of permutations $P$ of the set $\{1,\ldots,n\}$, there exists a closed infinite-dimensional Banach subspace of the space of $n$-linear forms on $\ell_{1}$ such that, for all nonzero elements $B$ of such a subspace, the Arens extension associated to the permutation $\sigma$ of $B$ is norm-attaining if and only if $\sigma$ is an element of $P$. We also study the structure of the set of norm-attaining $n$-linear forms on $c_{0}$.

#### Article information

Source
Banach J. Math. Anal., Volume 11, Number 1 (2017), 90-107.

Dates
Accepted: 25 February 2016
First available in Project Euclid: 10 November 2016

https://projecteuclid.org/euclid.bjma/1478746988

Digital Object Identifier
doi:10.1215/17358787-3750182

Mathematical Reviews number (MathSciNet)
MR3571146

Zentralblatt MATH identifier
1366.46032

#### Citation

Falcó, Javier; García, Domingo; Maestre, Manuel; Rueda, Pilar. Spaceability in norm-attaining sets. Banach J. Math. Anal. 11 (2017), no. 1, 90--107. doi:10.1215/17358787-3750182. https://projecteuclid.org/euclid.bjma/1478746988

#### References

• [1] M. D. Acosta, On multilinear mappings attaining their norms, Studia Math. 131 (1998), no. 2, 155–165.
• [2] M. D. Acosta, D. García, and M. Maestre, A multilinear Lindenstrauss theorem, J. Funct. Anal. 235 (2006), no. 1, 122–136.
• [3] R. Arens, The adjoint of a bilinear operation, Proc. Amer. Math. Soc. 2 (1951), 839–848.
• [4] R. M. Aron, L. Bernal-González, D. M. Pellegrino, and J. Seoane-Sepúlveda, Lineability: The Search for Linearity in Mathematics, Monogr. Res. Notes Math., CRC Press, Boca Raton, FL, 2015.
• [5] R. M. Aron and P. D. Berner, A Hahn–Banach extension theorem for analytic mappings, Bull. Soc. Math. France 106 (1978), no. 1, 3–24.
• [6] R. M. Aron, D. García, and M. Maestre, On norm attaining polynomials, Publ. Res. Inst. Math. Sci. 39 (2003), no. 1, 165–172.
• [7] R. M. Aron, V. I. Gurariy, and J. Seoane-Sepúlveda, Lineability and spaceability of sets of functions on $\mathbb{R}$, Proc. Amer. Math. Soc. 133 (2005), no. 3, 795–803.
• [8] R. M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, Bull. Amer. Math. Soc. 80 (1974), 1245–1249.
• [9] P. Bandyopadhyay and G. Godefroy, Linear structures in the set of norm attaining functionals on a Banach space, J. Convex Anal. 13 (2006), no. 3–4, 489–497.
• [10] L. Bernal-González, D. Pellegrino, and J. Seoane-Sepúlveda, Linear subsets of nonlinear sets in topological vector spaces, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 1, 71–130.
• [11] E. Bishop and R. R. Phelps, A proof that every Banach space is subreflexive, Bull. Amer. Math. Soc. 67 (1961), 97–98.
• [12] G. Botelho, D. Diniz, V. V. Fávaro, and D. Pellegrino Spaceability in Banach and quasi-Banach sequence spaces, Linear Algebra Appl. 434 (2011), no. 5, 1255–1260.
• [13] A. M. Davie and T. W. Gamelin, A theorem on polynomial-star approximation, Proc. Amer. Math. Soc. 106 (1989), no. 2, 351–356.
• [14] J. Diestel and J. J. Uhl, The Radon–Nikodym theorem for Banach space valued measures, Rocky Mountain J. Math. 6 (1976), no. 1, 1–46.
• [15] J. Falcó, D. García, M. Maestre, and P. Rueda, Norm Attaining Arens Extensions on $\ell_{1}$, Abstr. Appl. Anal. (2014), art. ID 315641.
• [16] V. I. Gurariy and L. Quarta, On lineability of sets of continuous functions, J. Math. Anal. Appl. 294 (2004), no. 1, 62–72.
• [17] J. Johnson and J. Wolfe, Norm attaining operators, Studia Math. 65 (1979), no. 1, 7–19.
• [18] J. Lindenstrauss, On operators which attain their norm, Israel J. Math. 1 (1963), 139–148.
• [19] M. Martín, Norm attaining compact operators, J. Funct. Anal. 267 (2014), no. 5, 1585–1592.
• [20] D. Pellegrino and E. Teixeira, Norm optimization problem for linear operators in classical Banach spaces, Bull. Braz. Math. Soc. (N.S.) 40 (2009), no. 3, 417–431.
• [21] V. Zizler, On some extremal problems in Banach spaces, Math. Scand. 32 (1973), 214–224.