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"The book presents a comprehensive exposition of extension results for maps between different geometric objects and of extension-trace results for smooth functions on subsets with no a priori differential structure (Whitney problems). The account covers development of the area from the initial classical works of the first half of the 20th century to the flourishing period of the last decade. Seemingly very specific these problems have been from the very beginning a powerful source of ideas, concepts and methods that essentially influenced and in some cases even transformed considerable areas of analysis. Aside from the material linked by the aforementioned problems the book also is unified by geometric analysis approach used in the proofs of basic results. This requires a variety of geometric tools from convex and combinatorial geometry to geometry of metric space theory to Riemannian and coarse geometry and more. The necessary facts are presented mostly with detailed proofs to make the book accessible to a wide audience."

"The book presents a comprehensive exposition of extension results for maps between different geometric objects and of extension-trace results for smooth functions on subsets with no a priori differential structure (Whitney problems). The account covers development of the area from the initial classical works of the first half of the 20th century to the flourishing period of the last decade. Seemingly very specific these problems have been from the very beginning a powerful source of ideas, concepts and methods that essentially influenced and in some cases even transformed considerable areas of analysis. Aside from the material linked by the aforementioned problems the book also is unified by geometric analysis approach used in the proofs of basic results. This requires a variety of geometric tools from convex and combinatorial geometry to geometry of metric space theory to Riemannian and coarse geometry and more. The necessary facts are presented mostly with detailed proofs to make the book accessible to a wide audience."

"The purpose of this book is to present some basic results in analysis that canbe used to solve various problems, so it may serve as a kind of toolbox. We donot intend to give a complete panorama of the field. In particular, we concentrateon real analysis and leave complex analysis aside; however we consider complexfunctions now and then. Although we cannot claim that this volume is devoted toapplied analysis, our choice of topics is driven by our wish to present results thatcan be applied to concrete problems. It is in this sense that this book could be called“Motivated Analysis”. This expression is due to J.-P. Aubin and means that we shallgather some results of analysis that may be useful in applications, even if the natureof these applications is not the focus of the text. Thus, our route is not the choice ofsome important problems that occur in applications, as in [22, 62, 87, 88, 101, 129,130, 140, 178, 191, 236, 241, 245], but neither is it the panorama of deep advancesin mathematical concepts presented in [100] and [101]."