Besides the very obvious change from German to English, the second edition of this book contains many additions as weil as a great many other changes. It might even be called a new book altogether were it not for the fact that the essential character of the book has remained the same; in other words, the entire presentation continues to be based on an axiomatic treatment of linear spaces. In this second edition, the thorough-going restriction to linear spaces of finite dimension has been removed. Another complete change is the restriction to linear spaces with real or complex coefficients, thereby removing a number of relatively involved discussions which did not really contribute substantially to the subject. On p.6 there is a list of those chapters in which the presentation can be transferred directly to spaces over an arbitrary coefficient field. Chapter I deals with the general properties of a linear space. Those concepts which are only valid for finitely many dimensions are discussed in a special paragraph. Chapter 11 now covers only linear transformations while the treatยญ ment of matrices has been delegated to a new chapter, chapter 111. The discussion of dual spaces has been changed; dual spaces are now introยญ duced abstractly and the connection with the space of linear functions is not established untillater
CONTENT
I. Linear spaces -- II. Linear transformations -- III. Matrices -- IV. Determinants -- V. Oriented linear spaces -- VI. Multilinear mappings -- VII. Tensor-algebra -- VIII. Exterior algebra -- IX. Duality in exterior algebra -- X. Inner product spaces -- XI. Linear mappings of inner product spaces -- XII. Symmetric bilinear functions -- XIII. Quadrics -- XIV. Unitary spaces -- XV. Invariant subspaces
