Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

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The Convenient Setting of Global Analysis

Overview.- An Overview of Empirical Processes.- Overview of Semiparametric Inference.- Case Studies I.- Empirical Processes.- to Empirical Processes.- Preliminaries for Empirical Processes.- Stochastic Convergence.- Empirical Process Methods.- Entropy Calculations.- Bootstrapping Empirical Processes.- Additional Empirical Process Results.- The Functional Delta Method.- Z-Estimators.- M-Estimators.- Case Studies II.- Semiparametric Inference.- to Semiparametric Inference.- Preliminaries for Semiparametric Inference.- Semiparametric Models and Efficiency.- Efficient Inference for Finite-Dimensional Parameters.- Efficient Inference for Infinite-Dimensional Parameters.- Semiparametric M-Estimation.- Case Studies III.

Introduction to empirical processes and semiparametric inference

In recent years, there has been an exponential growth in the number of complex documents and texts that require a deeper understanding of machine learning methods to be able to accurately classify texts in many applications. Many machine learning approaches have achieved surpassing results in natural language processing. The success of these learning algorithms relies on their capacity to understand complex models and non-linear relationships within data. However, finding suitable structures, architectures, and techniques for text classification is a challenge for researchers. In this paper, a brief overview of text classification algorithms is discussed. This overview covers different text feature extractions, dimensionality reduction methods, existing algorithms and techniques, and evaluations methods. Finally, the limitations of each technique and their application in real-world problems are discussed.

Text Classification Algorithms: A Survey

In recent years, there has been an exponential growth in the number of complex documents and texts that require a deeper understanding of machine learning methods to be able to accurately classify texts in many applications. Many machine learning approaches have achieved surpassing results in natural language processing. The success of these learning algorithms relies on their capacity to understand complex models and non-linear relationships within data. However, finding suitable structures, architectures, and techniques for text classification is a challenge for researchers. In this paper, a brief overview of text classification algorithms is discussed. This overview covers different text feature extractions, dimensionality reduction methods, existing algorithms and techniques, and evaluations methods. Finally, the limitations of each technique and their application in the real-world problem are discussed.

Preface * 1 Basic Concepts in Banach Spaces * 2 Hahn-Banach and Banach Open Mapping Theorems * 3 Weak Topologies * 4 Locally Convex Spaces * 5 Structure of Banach Spaces * 6 Schauder Bases * 7 Compact Operators on Banach Spaces * 8 Differentiability of Norms * 9 Uniform Convexity * 10 Smoothness and Structure * 11 Weakly Compactly Generated Spaces * 12 Topics in Weak Toplogy * References * Index

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Functional Analysis and Infinite-Dimensional Geometry

Classical Banach Spaces I: Sequence Spaces

This chapter explains the basic concepts in the geometry of Banach spaces. Banach spaces will have either real or complex scalars. When the scalar field matters, the scalar field is mentioned explicitly, but in the notation for special spaces the scalars are not specified. Operators between Banach spaces are bounded and linear. An invertible operator T is called an “isomorphism.” Two norms on a vector space are called “equivalent” if the identity operator on X (with the two given norms) is an isomorphism. A projection is an idempotent operator. A subspace Z of X is said to be complemented if there is a projection from X onto Z . Probability theory has had a profound impact on Banach space theory. Consequently, it is used in places probabilistic terminology. The modern interpolation theory and a few applications to Banach space theory are also discussed in this chapter.

Chapter 1 - Basic Concepts in the Geometry of Banach Spaces

It is proved that ifY ⊂X are metric spaces withY havingn≧2 points then any mapf fromY into a Banach spaceZ can be extended to a map
 
$$\hat f$$

 fromX intoZ so that
 
$$\left\| {\hat f} \right\|_{lip} \leqq c log n\left\| f \right\|_{lip} $$

 wherec is an absolute constant. A related result is obtained for the case whereX is assumed to be a finite-dimensional normed space andY is an arbitrary subset ofX.

Extensions of lipschitz maps into Banach spaces

A simple example is given of a non WCG space whose dual is a WCG space with an unconditional basis. It is proved that ifX* is WCG andX is a subspace of a WCG space thenX itself is WCG.

Some remarks on weakly compactly generated banach spaces

It is proved that for every reflexive Orlicz spaceX there is a functionn(k,e) so that wheneverE is ak-dimensional subspace ofX there exists an operatorT: X→X such thatT
 1E=identity, ‖T‖≦1+e and dimTX≦n(k,e). Some general facts concerning the uniform approximation property are also presented.

Joram Lindenstrauss

Papers

Classical Banach Spaces I: Sequence Spaces

Chapter 1 - Basic Concepts in the Geometry of Banach Spaces

Extensions of lipschitz maps into Banach spaces

Some remarks on weakly compactly generated banach spaces

The uniform approximation property in Orlicz spaces