# Manifolds, Tensor Analysis, and Applications by Ralph Abraham

By Ralph Abraham

The objective of this e-book is to supply middle fabric in nonlinear research for mathematicians, physicists, engineers, and mathematical biologists. the most target is to supply a operating wisdom of manifolds, dynamical structures, tensors, and differential types. a few functions to Hamiltonian mechanics, fluid me chanics, electromagnetism, plasma dynamics and keep watch over thcory arc given in bankruptcy eight, utilizing either invariant and index notation. the present version of the ebook doesn't care for Riemannian geometry in a lot element, and it doesn't deal with Lie teams, significant bundles, or Morse concept. a few of this is often deliberate for a next version. in the meantime, the authors will make on hand to readers supplementary chapters on Lie teams and Differential Topology and invite reviews at the book's contents and improvement. in the course of the textual content supplementary issues are given, marked with the symbols ~ and {l:;J. This gadget allows the reader to bypass numerous themes with out anxious the most circulation of the textual content. a few of these supply extra history fabric meant for completeness, to lessen the need of consulting too many open air references. We deal with finite and infinite-dimensional manifolds concurrently. this can be partially for potency of exposition. with no complicated purposes, utilizing manifolds of mappings, the examine of infinite-dimensional manifolds may be not easy to motivate.

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Proof Let e l , ... , en denote a basis of E, where n is the dimension of E. (i) A nonn on E is given, for example, by n lie II = Llail, i=1 where n e = L aiei i=1 (ii) Let II· II' be any other nonn on E. 1 Banach Spaces 47 is continuous with respect to the III· III -nonn on en. Since the set S = {x E en 1111 x 111= 1} is closed and bounded, it is compact. The restriction of this map to S is a continuous, strictly positive function, so it attains its minimum MI and maximum M2 on S; that is, for all (x I, ...

1 Definition A norm on a real (complex) vector space E is a mapping from E into the real numbers. II· II ; E ~ 1R; e ~ II e II. such that N1 II e II ~ 0 for all e E E and II e II =0 iff e =0 (positive definiteness); N2 II Ae II =I AI II e IIfor all e E E and A E 1R (homogeneity); N3 II e 1 + e 2 II :5 II e 1 II + II ~ II for all e 1• e2 E E (triangle inequality). 1 Banach Spaces The pair (E, II . II) is sometimes called a normed space. , II . liE' II . Jor the norm. The triangle inequality N3 has the following important consequence: which is proved in the following way: II e211 II e\ + (e2 - e\) II :5; II e\ II + II e\ - e211 , II e\ II = II e2 + (e\ - e2) II :5; II e211 + II e\ - e 2 II , = so that both II e211-11 e\ II and II e\ 11- II ~ II are smaller than or equal to II e\ - ~ II.

Show that for any continuous maps f, g : X -+ Y, the set {x E X I f(x) = g(x)} is closed. ) Thus, if f(x) = g(x) at all points of a dense subset of X, then f= g. 4D Define a topological manifold to be a space locally homeomorphic to JRn. Find a topological manifold that is not Hausdorff. 4E Show that a mapping