By Terence Tao

ISBN-10: 8185931631

ISBN-13: 9788185931630

** Read or Download Analysis II (Texts and Readings in Mathematics, No. 38) (Volume 2) PDF**

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

**Sample text**

13. Let E and F be two compact subsets of R (with the standard metric d(x, y) = lx- yl). Show that the Cartesian product Ex F := {(x,y): x E E,y E F} is a compact subset of R 2 (with the Euclidean metric d12). 14. Let (X, d) be a metric space, let E be a non-empty compact subset of X, and let Xo be a point in X. , x is the closest point in E to xo. (Hint: let R be the quantity R := inf{ d(xo, y) : y E E}. 15. Let (X, d) be a compact metric space. Suppose that (Ka)aEI is a collection of closed sets in X with the property that any finite subcollection of these sets necessarily has non-empty intersection, thus naEF Ka =/; 0 for all finite F ~I.

8. Iff: X~ R is a continuous function, then the function j2 : X ~ R defined by j2(x) := f(x) 2 is automatically continuous also. This is because we have f 2 = 9 o f, where 9 : R ~ R is the squaring function 9(x) := x 2 , and 9 is a continuous function. 1. 4. 4. 2. 5. 3. 7. 4. Give an example of functions such that f :R --t Rand g : R (a) f is not continuous, but g and go fare continuous; f and g o f are continuous; (c) f and g are not continuous, but go f is continuous. 2. 7. 5. Let (X, d) be a metric space, and let (E, diExE) be a subspace of (X, d).

6. 6 is indeed a metric space. 1. 7. 1. 7 is indeed a metric space. 8. 1). (For the first inequality, square both sides. 5). 9. 9 is indeed a metric space. 10. 2). 11. 11 is indeed a metric space. 12. 18. 13. 19. 14. 20. 15. Let 00 X:= {(an):=o: L lanl < oo} n=O be the space of absolutely convergent sequences. Define the l 1 and metrics on this space by 00 dtl((an):=O• (bn):=o) := L n=O ian- bnl; zoo 400 12. Metric spaces Show that these are both metrics on X, but show that there exist sequences x< 1>, x< 2 >, ...