The main algebraic structure that will be involved with the subject of this book is that of a “linear space” (or “vector space”). A linear space is a set endowed with an extra structure in addition to its set-theoretic structure (i.e., an extra structure that goes beyond the notions of functions and ordering, for instance). Roughly speaking, linear spaces are sets where two operations, called “addition” and “scalar multiplication”, are properly defined so that we can refer to the “sum” of two points in a linear space, as well as to the “product” of a point in it by a “scalar”. Although the reader is supposed to have already had some contact with linear algebra and, in particular, with “finite-dimensional vector spaces”, we shall proceed from the very beginning. Our approach avoids the parochially “finite-dimensional” constructions (whenever this is possible), and focuses either on general results that do not depend on the “dimensionality” of the linear space, or on abstract “infinite-dimensional” linear spaces.
KeywordsLinear Space Linear Transformation Algebraic Structure Binary Operation Scalar Multiplication
Unable to display preview. Download preview PDF.