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

By Terence Tao

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By Terence Tao

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

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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 >, ... , sequences of sequences) which are convergent with respect to the d1oo metric but not with respect to the d11 metric. Conversely, show that any sequence which converges in the dp metric automatically converges in the d1oo metric.

5. Let (X, d) be a compact metric space. Then (X, d) is both complete and bounded. Proof. 2. 6 (Compact sets are closed and bounded). Let (X, d) be a metric space, and let Y be a compact subset of X. Then Y is closed and bounded. 414 12. Metric spaces The other half of the Heine-Borel theorem is true in Euclidean spaces: Theorem 12. 5. 7 (Heine-Borel theorem). Let (Rn, d) be a Euclidean space with either the Euclidean metric, the taxicab metric, or the sup norm metric. Let E be a subset of R n.

In particular it is not possible for 402 12. Metric spaces x 0 to simultaneously be an interior and an exterior point of E. If x 0 is a boundary point of E, then it could be an element of E, but it could also not lie in E; we give some examples below. 7. We work on the real lineR with the standard metric d. Let E be the half-open intervalE= [1, 2). 5 which lies in E. 1)) centered at 3 which is disjoint from E. The points 1 and 2 however, are neither interior points nor exterior points of E, and are thus boundary points of E.

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