By Terence Tao

This is often half considered one of a two-volume creation to actual research and is meant for honours undergraduates, who've already been uncovered to calculus. The emphasis is on rigour and on foundations. the cloth begins on the very starting - the development of quantity platforms and set concept, then is going directly to the fundamentals of study (limits, sequence, continuity, differentiation, Riemann integration), via to energy sequence, numerous variable calculus and Fourier research, and at last to the Lebesgue necessary. those are nearly totally set within the concrete environment of the genuine line and Euclidean areas, even if there's a few fabric on summary metric and topological areas. There are appendices on mathematical common sense and the decimal process. the complete textual content (omitting a few much less valuable themes) might be taught in quarters of twenty-five to thirty lectures each one. The direction fabric is deeply intertwined with the routines, because it is meant that the coed actively research the cloth (and perform considering and writing carefully) by means of proving numerous of the main ends up in the speculation. the second one version has been commonly revised and up-to-date.

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**Example text**

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. Then E is compact if and only if it is closed and bounded.

Then f (E) is also connected. 4. 1. 4. 7 (Intermediate value theorem). Let f : X ~ R be a continuous map from one metric space (X, dx) to the real line. Let E be any connected subset of X, and let a, b be any two elements of E. , either f( a) ::; y ::; f(b) or f( a) ;:::: y ;:::: f(b ). Then there exists c E E such that f(c) = y. 5. 1. Let (X, ddisc) be a metric space with the discrete metric. Let E be a subset of X which contains at least two elements. Show that E is disconnected. 2. Let I: X-+ Y be a function from a connected metric space (X, d) to a metric space (Y, ddisc) with the discrete metric.

0 This theorem has an important consequence. 5 the notion of a function f : X ~ R attaining a maximum or minimum at a point. 6. 2 (Maximum principle). Let (X, d) be a compact metric space, and let f : X ~ R be a continuous function. Then f is bounded. Furthermore, f attains its maximum at some point Xmax E X, and also attains its minimum at some point Xmin EX. 2. 3. 1, this principle can fail if X is not compact. 6. 7. Another advantage of continuous functions on compact sets is that they are uniformly continuous.

### Analysis II (Texts and Readings in Mathematics) by Terence Tao

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