doc. RNDr. Jaroslav Tišer, CSc.

Existing negative results on invalidity of analogues of classical Density and Differentiation Theorems in infinite-dimensional spaces are considerably strengthened by a construction of a Gaussian measure gamma on a separable Hilbert space H for which the Density Theorem fails uniformly, i.e., there is a set M subset of H of positive gamma-measure such that

We introduce a criterion for a set to be Gamma-null. Using it we give a shorter proof of the result that the set of points where a continuous convex function on a separable Asplund space is not Frechet differentiable is Gamma-null. Our criterion also implies a new result about Gateaux (and Hadamard) differentiability of quasiconvex functions.

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.

This book compares the two computer algebra programs, Maple and Mathematica used by students, mathematicians, scientists, and engineers.

The proof of differentiability of Lipschitz function is done by using a new variational principle.

A new role of sigma ideal of porous sets is investigated.

Frechet Differentiability of Lipschitz maps and Porous Sets in Banach Spaces

In every Banach space there is a sigma porous set such that the complement is zero on every smooth curve.

There is Lipschitz function in the plane such that the set of points where the derivative exist is uniformly purely unrectifiable.