Analysis II by Terence Tao
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Questions and Answers

What is the distance function between two real numbers x and y?

d(x, y) := |x - y|

A sequence (xn) converges to a limit x if and only if limn→∞ d(xn, x) = 0.

True

What does it mean when a sequence x(n) converges to x with respect to a particular metric?

The distance between x(n) and x approaches zero as n approaches infinity.

In the example provided, what point does the sequence (1/n, 1/n) converge to in the Euclidean space R^2?

<p>(0, 0)</p> Signup and view all the answers

Which of the following metrics was mentioned as having the sequence (x(n)) converge to (0, 0)?

<p>Euclidean metric (dl2)</p> Signup and view all the answers

A sequence can converge to different points depending on the metric used.

<p>True</p> Signup and view all the answers

Who is the Advisory Editor of the Texts and Readings in Mathematics series?

<p>C.S.Seshadri</p> Signup and view all the answers

What kind of content does the Texts and Readings in Mathematics series publish?

<p>High-quality textbooks</p> Signup and view all the answers

The Texts and Readings in Mathematics series is co-published with Hindustan Book Agency in print format only.

<p>False</p> Signup and view all the answers

Terence Tao is affiliated with the Department of Mathematics at the University of California, ______.

<p>Los Angeles</p> Signup and view all the answers

What is the definition of a metric ball in a metric space?

<p>The ball B(X,d) (x0 , r) in X, centered at x0, and with radius r, in the metric d, is the set {x ∈ X : d(x, x0) &lt; r}</p> Signup and view all the answers

In R2 with the Euclidean metric dl2, what shape is the ball B(R2,dl2) ((0, 0), 1)?

<p>Open disc</p> Signup and view all the answers

In a metric space, a point that is an interior point of a set must also be an element of that set.

<p>True</p> Signup and view all the answers

The set of all __ points of a set E is called the interior of E.

<p>interior</p> Signup and view all the answers

Match the following: Which point is neither an interior nor an exterior point of a set E?

<p>a = Boundary point b = Exterior point c = Interior point</p> Signup and view all the answers

What is the definition of a relatively open set with respect to Y?

<p>E is relatively open with respect to Y if it is open in the metric subspace (Y, d|Y ×Y).</p> Signup and view all the answers

Provide the definition of a Cauchy sequence in a metric space.

<p>A sequence (x(n))∞n=m in a metric space (X, d) is a Cauchy sequence if for every ε &gt; 0, there exists an N ≥ m such that d(x(j), x(k)) &lt; ε for all j, k ≥ N.</p> Signup and view all the answers

Every Cauchy sequence in a metric space will converge to a limit point.

<p>False</p> Signup and view all the answers

Which of the following statements about compact metric spaces is true?

<p>Closed and bounded sets in compact spaces are compact.</p> Signup and view all the answers

A metric space (X, d) is said to be __________ if every sequence in (X, d) has at least one convergent subsequence.

<p>compact</p> Signup and view all the answers

What does the Heine-Borel theorem state about compact sets in Euclidean spaces?

<p>Heine-Borel theorem states that a subset in a Euclidean space is compact if and only if it is closed and bounded.</p> Signup and view all the answers

Which properties are necessary for a set to be considered compact in a metric space? (Select all that apply)

<p>Bounded</p> Signup and view all the answers

The Heine-Borel theorem is not true for more general metrics, such as the integers Z with the _ metric.

<p>discrete</p> Signup and view all the answers

Every open cover of a compact set must have a finite subcover.

<p>True</p> Signup and view all the answers

Match the following properties with compact sets: (a) If Y is a compact subset of X, and Z ⊆ Y, then Z is compact if and only if Z is closed. (b) If Y1,...,Yn are a finite collection of compact subsets of X, then their union Y1 ∪...∪ Yn is also compact. (c) Every finite subset of X is compact.

<p>a = Closed b = Union of compact sets is compact c = Every finite subset is compact</p> Signup and view all the answers

Define what it means for a function to be continuous at a point in a metric space.

<p>A function f is continuous at a point x0 in a metric space if for every ε &gt; 0, there exists a δ &gt; 0 such that dY(f(x), f(x0)) &lt; ε whenever dX(x, x0) &lt; δ.</p> Signup and view all the answers

Which statement is equivalent to stating that a function is continuous at a given point in a metric space?

<p>For every open set in the output space containing the function value, there exists an open set in the input space containing the point.</p> Signup and view all the answers

Continuous functions preserve convergence. Is this statement true or false?

<p>True</p> Signup and view all the answers

______ functions are also sometimes called continuous maps.

<p>Continuous</p> Signup and view all the answers

What is a metric space?

<p>A space with a concept of distance between points.</p> Signup and view all the answers

Define the Euclidean metric (l2 metric).

<p>The Euclidean metric (l2 metric) is defined as the square root of the sum of squares of the differences between corresponding coordinates of two points.</p> Signup and view all the answers

In the ___, all points are equally far apart.

<p>discrete metric</p> Signup and view all the answers

For any distinct points x, y in a metric space, the distance function is always greater than 0.

<p>True</p> Signup and view all the answers

Study Notes

Texts and Readings in Mathematics

  • The Texts and Readings in Mathematics series publishes high-quality textbooks, research-level monographs, lecture notes, and contributed volumes.
  • 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.

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Analysis II is a mathematics textbook by Terence Tao, part of the Texts and Readings in Mathematics series, suitable for undergraduate and graduate students.

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