Abstract

We exhibit new classes of Banach spaces that have the strong-1121\frac{1}{2}-ball property and the 1121\frac{1}{2}-ball property by considering direct-sums of Banach spaces. We introduce the notion of sectional strong-1121\frac{1}{2}-ball property and show that in c0c_0-direct sum of reflexive spaces, proximinal and factor reflexive spaces with the sectional strong-1i21\frac{i}{2}-ball property have the strong-1121\frac{1}{2}-ball property. We give examples of proximinal hyperplanes in c0c_0 that fail the 1121\frac{1}{2}-ball property and show that this property is in general, not preserved under finite intersections or sums. We show that the range of a bi-contractive projection in ℓ∞\ell^{\infty} has the strong-1121\frac{1}{2}-ball property. For a separable subspace Y⊂XY \subset X with the strong-1121\frac{1}{2}-ball property and for any positive, σ\sigma-finite, non-atomic measure space (Ω,A,μ)(\Omega, {\mathcal A}, \mu), we show that L1(μ,Y)L^1(\mu,Y) has the strong-1121\frac{1}{2}-ball property in L1(μ,X)L^1(\mu,X). We show that for any compact set Ω\Omega and Y⊂XY \subset X with the 1121\frac{1}{2}-ball property, C(Ω,Y)C(\Omega,Y) has the 1121\frac{1}{2}-ball property in C(Ω,X)C(\Omega,X).

Author information

Contact details are reproduced from the original publication and may be historical.

T. S. S. R. K. Rao

Indian Statistical Institute, R. V. College P.O., Bangalore 560059, India

tss@isibang.ac.in

Keywords

  • One and half ball property
  • spaces of vector-valued functions

Mathematics Subject Classification

41A65,46B20; 41A50

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Suggested citation

T. S. S. R. K. Rao. “The One and Half Ball Property in Spaces of Vector-Valued Functions.” Journal of Convex Analysis 20 (2013), No. 1, 13–23. https://doi.org/10.68381/jca20002

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Published by Heldermann Verlag, 2013. Rights now held by Banach Press.