Abstract

A well known criterion of Šmulyan states that the norm ∥⋅∥\|\cdot\| of a real Banach space XX is Gâteaux differentiable at x∈Xx\in X if and only if there is x∗∈SX∗x^*\in S_{X^*} which is w∗w^*-exposed by xx in BX∗B_{X^*} and that the norm is Fréchet differentiable at xx if and only if there is x∗∈SX∗x^*\in S_{X^*} which is w∗w^*-strongly exposed in BX∗B_{X^*} by xx. We show that in this criterion BX∗B_{X^*} can be replaced by a convenient smaller set, and we apply this extended criterion to characterize the points of Gâteaux and Fréchet differentiability of the norm in epsilon products of Banach spaces, extending previous work of Heinrich. As a consequence we get some results of smoothness of the norm in some Banach spaces of continuous and harmonic vector valued functions.

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Enrique Jordá

Dep. de Matemática Aplicada, Universidad Politécnica de Valencia, Plaza Ferrándiz y Carbonell 2, 03801 Alcoy, Spain

ejorda@mat.upv.es

Ana María Zarco

Dep. de Matemática Aplicada, Universidad Politécnica de Valencia, Plaza Ferrándiz y Carbonell 2, 03801 Alcoy, Spain

anzargar@upv.es

Keywords

  • Banach spaces
  • Frechet and Gateaux differentiability
  • epsilon products

Mathematics Subject Classification

46B20, 46B50

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Suggested citation

E. Jordá, A. M. Zarco. “Smoothness in some Banach Spaces of Operators and Vector Valued Functions.” Journal of Convex Analysis 26 (2019), No. 2, 515–526. https://doi.org/10.68381/jca26025

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Published by Heldermann Verlag, 2019. Rights now held by Banach Press LLC.