Terence Tao's Analysis II (Texts and Readings in Mathematics, No. 38) PDF

By Terence Tao

ISBN-10: 8185931631

ISBN-13: 9788185931630

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Extra info for Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2)

Example text

If xo E E and f is continuous at x 0 , show that /IE is also continuous at xo. (Is the converse of this statement true? IE is continuous. Thus restriction of the domain of a function does not destroy continuity. 7. Let f : X ---+ Y be a function from one metric space (X, dx) to another (Y, dy ). Suppose that the image f(X) of X is contained in some subset E C Y of Y. Let g : X ---+ E be the function which is the same as f but with the range restricted from Y toE, thus g(x) = f(x) for all x EX. We give E the metric dyiExE induced from Y.

IE is continuous. Thus restriction of the domain of a function does not destroy continuity. 7. Let f : X ---+ Y be a function from one metric space (X, dx) to another (Y, dy ). Suppose that the image f(X) of X is contained in some subset E C Y of Y. Let g : X ---+ E be the function which is the same as f but with the range restricted from Y toE, thus g(x) = f(x) for all x EX. We give E the metric dyiExE induced from Y. Show that for any x 0 E X, that f is continuous at x 0 if and only if g is continuous at xo.

Relative topology that E is relatively open with respect to Y if it is open in the metric subspace (Y, diYxY). Similarly, we say that E is relatively closed with respect to Y if it is closed in the metric space (Y, diYxY). The relationship between open (or closed) sets in X, and relatively open (or relatively closed) sets in Y, is the following. 4. Let (X, d) be a metric space, let Y be a subset of X, and let E be a subset of Y. (a) E is relatively open with respect to Y if and only if E for some set V <;;;; X which is open in X.

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Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2) by Terence Tao


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