Tensor spaces and numerical tensor calculus Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd where nd exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail a particular tensor calculus is needed to treat such problems. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry approximation of multivariate functions solution of partial differential equations for example with stochastic coefficients and more. In addition to containing corrections of the unavoidable misprints this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high dimensional data. It also touches problems related to Geometric Algebra. Part I: Algebraic Tensors 1 Introduction 2 Matrix Tools 3 Algebraic Foundations of Tensor Spaces Part II: Functional Analysis of Tensor Spaces 4 Banach Tensor Spaces 5 General Techniques 6 Minimal Subspaces Part III: Numerical Treatment 7 r Term Representation 8 Tensor Subspace Represenation 9 r Term Approximation 10 Tensor Subspace Approximation 11 Hierarchical Tensor Representation 12 Matrix Product Systems 13 Tensor Operations 14 Tensorisation 15 Multivariate Cross Approximation 16 Applications to Elliptic Partial Differential Equations 17 Miscellaneous Topics.