By Werner Hildbert Greub, Stephen Halperin, Ray Vanstone

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**Example text**

Hence, for each h E E, the maps y Ad is smooth. I-+ Ad y ( h ) are smooth. D.

In this case the derivative of v' is denoted by v";it is a map y": U - L(E;L(E;F))= L(E,E;F). , E;F). k terms For each a E U , v ( k ) ( a )is a symmetric k-linear map of E x x E into F. If all derivatives of cp exist, v is called infinitely dzyerentiable, or smooth. A smooth map v: U -+ V between open subsets U C E and V C F is called a di@omorphism if it has a smooth inverse. Assume now that y : U + F is a map with a continuous derivative such that for some point a E U y'(a): E 5 -F is a linear isomorphism.

Relations (0) and (1) imply that exp u is an automorphism with (exp u)-' = exp(-u). In particular, if u is self-adjoint, then so is exp u and if u is skew (resp. Hermitian skew), then exp u is a proper rotation (resp. unitary transformation) of E. 1 1. General topology. We shall assume the basics of point set topology: manipulation with open sets and closed sets, compactness, Hausdorff spaces, locally compact spaces, second countable spaces, connectedness, paracompact spaces, normality, open coverings, shrinking of an open covering, etc.