The book introduces classical inequalities in vector and functional spaces with applications to probability. It develops new analytical inequalities, with sharper bounds and generalizations to the sum or the supremum of random variables, to martingales, to transformed Brownian motions and diffusions, to Markov and point processes, renewal, branching and shock processes.In this third edition, the inequalities for martingales are presented in two chapters for discrete and time-continuous local martingales with new results for the bound of the norms of a martingale by the norms of the predictable processes of its quadratic variations, for the norms of their supremum and their p-variations. More inequalities are also covered for the tail probabilities of Gaussian processes and for spatial processes.This book is well-suited for undergraduate and graduate students as well as researchers in theoretical and applied mathematics.Key Features: oThe book is complementary to the classical courses on vector and functional Hilbert spaces, integration theory and probability which focus on inequalitiesoThe preliminary chapter introduces most classical inequalities which are generalized to more complex models with detailed proofsoIt covers different domains from existing books and contains many new resultsoIt gathers inequalities from several domains of mathematics which are not generally presented together, in a unified approachoThe topics include many stochastic processes with specific inequalities and the basis of the stochastic calculus is developed in numerous applications

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