By Tsoy-Wo Ma

This ebook presents an easy advent to the classical research of normed areas, paying designated recognition to non-linear subject matters corresponding to fastened issues, calculus and traditional differential equations. it really is aimed toward novices who are looking to get during the easy fabric once attainable after which movement directly to do their very own study instantly. It assumes simply common wisdom in finite-dimensional linear algebra, basic calculus and easy advanced research. because the remedy is self-contained with adequate info, even an undergraduate with mathematical adulthood should not have any challenge operating via it on my own. a number of chapters should be built-in into components of a grasp measure programme by way of direction paintings prepared through any local collage. constrained to finite-dimensional areas instead of normed areas, chosen chapters can be utilized for a direction in complicated calculus. Engineers and physicists may well locate this e-book a convenient reference in classical research

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This completes the proof. 9. Exercise Show that the continuous function j on (0,1) has no continuous extension over [0,1]. 10. Exercise Prove that the projections from a product metric space XxY onto its factor spaces X, Y are uniformly continuous. 1. Let A" be a metric space. A pair {A, B) of non-empty subsets of X is called a disconnection if A n ~B = 9 and B n ~A = 0. A subset A f of A" is said to be disconnected if there is a disconnection (A, B) such that Af = A U B. A subset of A" is said to be connected if it is not disconnected.

8. e. A C U > i Vi & *8 open. Then there are closed subsets Bj of A such that A = U B^ and B j C Vj for each 1 < j < n. w 9 X 6 f c l Proof. It suffices to prove the case for n = 2. Let f/, V be open sets such that A c f / n V . Let A f = A \ V and N = A\U. Then both Af, JV are closed sets and they are disjoint. There is a continuous function / on X such that / ( A f ) = 0 and f(N) = 1. Then both P = / ( - o o , j ) and Q = / " ' ( ^ c o ) are open sets containing Af, N respectively. Therefore E = A\Q and F = A \ P are closed subsets of A.

Consider any open ball B(j/,e). Theorem n Since e > 0 is • Let A" be a metric space. Then the following statements are equivalent. (a) X is complete. (b) Let An+i C A n be a decreasing sequence of non-empty closed sets. If their diameters tend to zero then fl™, A„ contains exactly one point. 3. Complete, Compact and Connected Sets 24 (c) A sequence {x„} in X satisfying x „ ) < 1/2" for all n , is convergent. Proof, (a =t- 6) For each n, take any x „ € A . Since diam(A ) -< 0, for every e > 0 there is p > 1 such that for every n > p, we have diam(A„) < e.