Analysis II by Terence Tao

Texts and Readings in Mathematics

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• The series is useful for undergraduate and graduate students of mathematics, research scholars, and teachers.

Analysis II

• Analysis II is a textbook written by Terence Tao, a mathematician at the University of California, Los Angeles.
• The book is a part of the Texts and Readings in Mathematics series.
• It is a co-publication with Hindustan Book Agency, New Delhi, India.

Editions

• The second and third editions of the book incorporate corrections and changes to the layout, page numbering, and indexing.
• The mathematical content remains the same as the first edition, and the two editions can be used interchangeably.

Preface to the Second and Third Editions

• The second and third editions include corrections reported by students and lecturers.
• The layout, page numbering, and indexing have been changed, and the book is now available in hardcover.
• The chapter and exercise numbering remains the same as the first edition.

Preface to the First Edition

• The book originated from the author's lecture notes on real analysis for undergraduate students at the University of California, Los Angeles.
• The author's approach to teaching real analysis was to start from the foundations of mathematics, including the definition of natural numbers, integers, and real numbers.
• The course covered the basics of set theory, including the uncountability of the real numbers.
• The author's goal was to help students understand the rigorous proofs and develop their mathematical skills.

Course Structure

• The course started with the basics of natural numbers, integers, and real numbers.
• The author covered the properties of these number systems, including the associative property of addition and the existence of rational numbers.
• The course then moved on to more advanced topics, including limits, continuity, and differentiability.
• The author emphasized the importance of rigorous proofs and encouraged students to write their own proofs.

Student Response

• Students found the material challenging but rewarding, as it helped them develop their mathematical skills.
• The author's approach helped students connect abstract mathematical concepts to their intuitive understanding of the world.

Exercises

• The book includes exercises that are essential to the course, including proving lemmas, propositions, and theorems.
• The author recommends that students do as many exercises as possible to truly understand the material.
• The exercises are listed at the end of each section, and the author suggests that students use them to solidify their understanding of the concepts.### Metric Spaces and Convergence
• A metric space is a set of objects (called points) with a concept of distance, which is a non-negative real number, between each pair of points.
• The distance function, also called a metric, must satisfy four axioms:
  • (a) For any x, d(x, x) = 0.
  • (b) For any distinct x, y, d(x, y) > 0.
  • (c) For any x, y, d(x, y) = d(y, x).
  • (d) For any x, y, z, d(x, z) ≤ d(x, y) + d(y, z).
• Examples of metric spaces:
  • The real line R, with the standard metric d(x, y) = |x - y|.
  • Euclidean spaces R^n, with the Euclidean metric d((x1, ..., xn), (y1, ..., yn)) = sqrt((x1 - y1)^2 + ... + (xn - yn)^2).
  • Taxicab metric (or l1 metric) d((x1, ..., xn), (y1, ..., yn)) = |x1 - y1| + ... + |xn - yn|.
  • Sup norm metric (or l∞ metric) d((x1, ..., xn), (y1, ..., yn)) = max(|x1 - y1|, ..., |xn - yn|).
  • Discrete metric d(x, y) = 0 if x = y, and 1 otherwise.
  • Geodesics (shortest curves) on the sphere {(x, y, z) ∈ R³: x² + y² + z² = 1}.
  • Shortest paths in a network of computers.

Convergence of Sequences in Metric Spaces

• A sequence (x(n))∞ n=m in a metric space (X, d) converges to a point x if and only if for every ε > 0, there exists an N ≥ m such that d(x(n), x) ≤ ε for all n ≥ N.
• Convergence is independent of the starting point m of the sequence.
• The convergence of a sequence can depend on the metric used.
• The Euclidean, taxi-cab, and sup norm metrics on R^n are equivalent, meaning that a sequence converges in one of these metrics if and only if it converges in the others.

Key Results

• Proposition 1.1.18: The equivalence of the Euclidean, taxi-cab, and sup norm metrics on R^n.Here are the study notes:

Metric Spaces

• A metric space is a set with a metric (distance function) that satisfies certain properties.

Convergence of Sequences

• A sequence converges to a point if and only if it is eventually constant in the discrete metric.
• Converging sequences can only converge to at most one point at a time (Uniqueness of Limits).
• The notation d - lim n→∞ x(n) = x or lim n→∞ x(n) = x is used to indicate that a sequence converges to a point.

Open and Closed Sets

• A set is open if it contains all its interior points.
• A set is closed if it contains all its boundary points.
• The interior, exterior, and boundary of a set can be defined using metric balls.
• The closure of a set is the set of all adherent points.
• A set is open if and only if it is equal to its interior.
• A set is closed if and only if it contains all its adherent points.

Properties of Open and Closed Sets

• The whole space and the empty set are both open and closed.
• A set is open if and only if its complement is closed.
• The intersection of open sets is open, and the union of closed sets is closed.
• The intersection of a finite collection of open sets is open, and the union of a finite collection of closed sets is closed.
• The union of an arbitrary collection of open sets is open, and the intersection of an arbitrary collection of closed sets is closed.

Relative Topology

• The same set can be open or closed depending on the ambient space.
• A set is relatively open with respect to a subset if it is open in the metric subspace.
• A set can be closed in a subspace but not in the ambient space.

I hope this helps! Let me know if you have any questions or need further clarification.