# Fr ´Echet Differentiability of Vector-Valued Functions

# Fr ´Echet Differentiability of Vector-Valued Functions

This chapter shows that if a Banach space with a Fréchet smooth norm is asymptotically smooth with modulus *o*(*t*ⁿ logⁿ⁻¹(1/*t*)) then every Lipschitz map of *X* to a space of dimension not exceeding *n* has many points of Fréchet differentiability. In particular, it proves that two real-valued Lipschitz functions on a Hilbert space have a common point of Fréchet differentiability. The chapter first presents the theorem whose assumptions hold for any space *X* with separable dual, includes the result that real-valued Lipschitz functions on such spaces have points of Fréchet differentiability, and takes into account the corresponding mean value estimate. The chapter then gives the estimate for a “regularity parameter” and reduces the theorem to a special case. Finally, it discusses simplifications of the arguments of the proof of the main result in some special situations.

*Keywords:*
separable dual, Banach space, Fréchet smooth norm, modulus, Lipschitz map, Fréchet differentiability, Lipschitz function, Hilbert space, mean value estimate, regularity parameter

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