"The book is an introduction to advanced analysis at the beginning graduate level that blends a modern presentation with concrete examples and applications, in particular in the areas of calculus of variations and partial differential equations. The book does not strive for abstraction for its own sake, but tries rather to impart a working knowledge of the key methods of contemporary analysis, in particular those that are also relevant for application in physics. It provides a streamlined and quick introduction to the fundamental concepts of Banach space and Lebesgue integration theory and the basic notions of the calculus of variations, including Sobolev space theory."--BOOK JACKET.Title Summary field provided by Blackwell North America, Inc. All Rights Reserved
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Limits and Continuity of Functions
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assume assumption Banach space bounded sequence called Cauchy sequence Cc(Rd choose closed consider const constant continuous functions continuously differentiable convergent subsequence converges pointwise converges uniformly convex corollary cube curve defined differentiable at xo differentiable function differential equations dimensional Dirichlet dominated convergence theorem equivalent Euler-Lagrange equations example exists f f(x)dx finite fn(x fn(x)dx follows directly function f function theorem Furthermore harmonic functions hence Hilbert space holds implies integrable function interval Laplace operator Lebesgue integral lemma Let f Let n)neN lim xn liminf limit linear map lower semicontinuous matrix maximum measurable functions metric space minimum n)neN converges normed vector space notations null function null set numbers obtain oo oo partially differentiable proof of theorem prove Rd be open Rd Rd satisfies sequence n)neN Sobolev Theory topology triangle inequality unique weak convergence weak derivative weak solution weakly