`Cartan's work provides a superb text for an undergraduate course in advanced calculus, but at the same time it furnishes the reader with an excellent foundation for global and nonlinear algebra.`—Mathematical Review `Brilliantly successful.`—Bulletin de l'Association des Professeurs de Mathematiques `The presentation is precise and detailed, the style lucid and almost conversational . . . clearly an outstanding text and work of reference.`—Annales Cartan's Formes Differentielles was first published in France in 1967. It was based on the world-famous teacher's experience at the Faculty of Sciences in Paris, where his reputation as an outstanding exponent of the Bourbaki school of mathematics was first established. Addressed to second- and third-year students of mathematics, the material skillfully spans the pure and applied branches in the familiar French manner, so that the applied aspects gain in rigor while the pure mathematics loses none of its dignity. This book is equally essential as a course text, as a work of reference, or simply as a brilliant mathematical exercise.
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1.1 Definition of an alternating multilinear mapping
Much of the following will be of interest purely from the point of view of that division of algebra which is concerned with the study of vector spaces over an arbitrary commutative field K (no restriction being placed on the characteristic of the field). However, we shall confine ourselves in exposition to normed vector spaces over the fields R or C: to illustrate, let us take R.
Suppose then that E, F are two normed vector spaces. We have already considered (Chap. 1, § 1.8, Differential Calculus) the normed vector space
p(E; F) of the p-linear continuous mappings Ep → F; for p = 1 we write for simplicity
(E; F); for p = 0 we define
0(E; F) = F. We also considered the vector subspace of
p(E; F) formed by the p-linear symmetric mappings. We shall now introduce a second subspace
of
p(E; F).
DEFINITION. A mapping f ∈
p(E; F) is called alternating if its value f(x1, ..., xp) is null whenever xi = xi+1 for at least one i (1 ≤ i < p). (We agree that for p = 1 every linear function E → F is an alternation.)
It is evident that the p-linear alternating mappings form a vector subspace of
p(E; F); we denote it by
Then
; by definition we put
The vector subspace
is closed in
P(E; F) ; indeed, suppose f ∈
p(E; F) is the limit of a sequence fn ∈
: then limn→∞ || f
fn || = 0; a fortiori, for x1, . . ., xp ∈ E, the limit of ƒn(x1, ..., xp) is ƒ(x1, ... , xp). Thus if xi = xi+1, it follows that
Q.E.D.
Before stating further properties of multilinear alternating mappings we require a few preliminaries concerning permutation groups.
