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This book makes a significant inroad into the unexpectedly difficult question of existence of Fréchet derivatives of Lipschitz maps of Banach spaces into higher dimensional spaces. Because the question turns out to be closely related to porous sets in Banach spaces, it provides a bridge between descriptive set theory and the classical topic of existence of derivatives of vector-valued Lipschitz functions. The topic is relevant to classical analysis and descriptive set theory on Banach spaces. The book opens several new research directions in this area of geometric nonlinear functional analysis. The new methods developed here include a game approach to perturbational variational principles that is of independent interest. Detailed explanation of the underlying ideas and motivation behind the proofs of the new results on Fréchet differentiability of vector-valued functions should make these arguments accessible to a wider audience. The most important special case of the differentiability results, that Lipschitz mappings from a Hilbert space into the plane have points of Fréchet differentiability, is given its own chapter with a proof that is independent of much of the work done to prove more general results. The book raises several open questions concerning its two main topics.

Banach spaces. --- Calculus of variations. --- Functional analysis. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Functions of complex variables --- Generalized spaces --- Topology --- Asplund space. --- Banach space. --- Borel sets. --- Euclidean space. --- Frechet differentiability. --- Fréchet derivative. --- Fréchet differentiability. --- Fréchet smooth norm. --- Gâteaux derivative. --- Gâteaux differentiability. --- Hilbert space. --- Lipschitz function. --- Lipschitz map. --- Radon-Nikodým property. --- asymptotic uniform smoothness. --- asymptotically smooth norm. --- asymptotically smooth space. --- bump. --- completeness. --- cone-monotone function. --- convex function. --- deformation. --- derivative. --- descriptive set theory. --- flat surface. --- higher dimensional space. --- infinite dimensional space. --- irregular behavior. --- irregularity point. --- linear operators. --- low Borel classes. --- lower semicontinuity. --- mean value estimate. --- modulus. --- multidimensional mean value. --- nonlinear functional analysis. --- nonseparable space. --- null sets. --- perturbation function. --- perturbation game. --- perturbation. --- porosity. --- porous sets. --- regular behavior. --- regular differentiability. --- regularity parameter. --- renorming. --- separable determination. --- separable dual. --- separable space. --- slice. --- smooth bump. --- subspace. --- tensor products. --- three-dimensional space. --- two-dimensional space. --- two-player game. --- variational principle. --- variational principles. --- Γ-null sets. --- ε-Fréchet derivative. --- ε-Fréchet differentiability. --- σ-porous sets.