By Terence Tao

This can be half considered one of a two-volume creation to genuine 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 research (limits, sequence, continuity, differentiation, Riemann integration), via to strength sequence, numerous variable calculus and Fourier research, and eventually to the Lebesgue necessary. those are nearly solely set within the concrete atmosphere 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 procedure. the full textual content (omitting a few much less principal subject matters) could be taught in quarters of twenty-five to thirty lectures each one. The direction fabric is deeply intertwined with the workouts, because it is meant that the coed actively study the fabric (and perform pondering and writing carefully) through proving numerous of the foremost ends up in the idea. the second one version has been broadly revised and up-to-date.

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**Example text**

8EnE = 0. If E contains some of its boundary points but not others, then it is neither open nor closed. 13. We work in the real line R with the standard metric d. The set (1, 2) does not contain either of its boundary points 1, 2 and is hence open. The set [1, 2] contains both of its boundary points 1, 2 and is hence closed. The set [1, 2) contains one of its boundary poi~ts 1, but does not contain the other boundary point 2, but not the other, so is neither open nor closed. 14. It is possible for a set to be simultaneously open and closed, if it has no boundary.

1. 2. 2. 3. Show that every uniformly continuous function is continuous, but give an example that shows that not every continuous function is uniformly continuous. 4. Let (X,dx), (Y,dy), (Z,dz) be metric spaces, and let f : X --+ Y and g : Y --+ Z be two uniformly continuous functions. Show that g o f : X --+ Z is also uniformly continuous. 5. Let (X, dx) be a metric space, and let f : X --+ R and g : X --+ R be uniformly continuous functions. Show that the direct sum fEB g: X--+ R 2 defined by fEB g(x) := (f(x),g(x)) is uniformly continuous.

1 (Continuous maps preserve compactness). Let f : X ~ Y be a continuous map from one metric space (X, dx) to another (Y, dy). Let K ~ X be any compact subset of X. Then the image f(K) := {f(x): x E K} of K is also compact. Proof. 1. 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.

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

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