math 142b homework solution

math 142b homework solution

Advanced Mathematics: Math 142B Homework Solutions

1. Introduction to Math 142B Homework

1. Introduction to Math 142B Homework.

Here you will find recollections, homework solutions, and other assorted materials from a graduate class: Math 142B, or Analysis II. These should be available to anyone who has taken an equivalent course. I will refer without proof to various results from this course. If you are uncertain about any of these results, I would like to invite you to post a comment on the relevant blog post, and I will be happy to answer. Most of the homework problems are taken from the monograph of Real Function Algebras written by Steven John Alpern.

This course is designed to follow upon the prerequisite requirements of Math 142A or Analysis I. I would like to remind the reader that it is important to remember that we are dealing with real-valued functions, unless we are specifically told otherwise. Since at present it seems that this will be a running example, we might ask several questions. What sorts of problems might we encounter in the study of a real-valued function? We are all familiar that R may not itself be convenient in which to work, but what sorts of spaces are available to us? What sorts of properties ought such spaces have, and how can they be classified? And so on.

2. Problem Solving Strategies and Techniques

1.2. (a) Prove that no function from the positive integers to themselves can take on each of the values 1966, 1967, 1968, 1970.

(b) Show that a function which takes on each of the values 1, 2, 3, 9, 10 must take on at least 30 values.

1.3. Let n be a natural number, and let A1, A2, …, An be different subsets of {1, 2, …, n} such that both A1 and A2 contain at least one element. Prove that there are two other disjoint subsets B1 and B2 of {1, 2, …, n} such that

B1 ∩ Aj = B2 ∩ Aj for j = 1, 2, …, n

=> B1 = A1 – A2 and B2 = A2 – A1

The problem is based on the following fact: in any game of chance, knowing the numbers on all but two of the balls drawn is exactly as good as knowing the numbers on all of them. At first, we compare the elements of A1 and A2. It seems quite reasonable that the numbers that do not appear in A1 are as important as the numbers that do. Indeed, we often utilize such information in solving various problems. Thus, it would be desirable to choose B1 and B2 in such a way that they are disjoint. These considerations should lead to the ideas that in general we should try inequalities and induction by n. Indeed, we apply the latter and try to find the answer in the case of n = 2.

3. Detailed Solutions and Explanations for Selected Problems

I. The number 2 is less than 3 and is therefore correct. Using this fact with Theorem 2.1, we get that Q is irrational. Recall from class that when Q is irrational, exponential and logarithmic functions are defined in terms of the integral power. Our strategy for constructing an ordered field that contains all rational numbers is to let IR be the set of all real-valued relations of rational numbers that have properties similar to those of irrational numbers. We then, in effect, replace the concepts of rational, real, and irrational by extensions of these relations, name their extensions rational, irrational, and real, relax the axioms of the ordered field by interpreting them in the context of these new relations, and reestablish the operations of addition and multiplication.

III. The only missing piece from the proof of the existence of a real number system is Proposition 2.1(e). Before we prove the result, we make two observations. We note first that Q has no smallest element. If d ≤ 0, then 2-21 = 1 + 1 ≤ d. Therefore 2 – 81 ≤ 0. This observation is also an observation of Proposition 2.3, and it is expressed schematically in Q as d ≤ 0 ≤ 2-21 ≤ 0 ≤ 21 ≤ d.

4. Additional Resources for Further Practice and Understanding

Many students find additional study, practice, and review to be quite beneficial. The following are great resources for this course. Other resources may be obtained from the Mathematics Department.

Related websites and web pages of discussion and information: A major site for main information and helpful links of clarification is maintained by the course coordinator. This will also direct you to other web pages giving information on each course. Continual watching of these sites is recommended.

Math 142B Discussion Board: The course coordinator has set up a discussion board for use exclusively for the students in Math 142B. This can include discussions on homework, how to solve various problems, etc. To access, moderators will periodically send the class an email with login instructions. You must enroll in order to access. If there is no moderator in a day or so, the Discussion Board can be accessed through information given on the course coordinator’s web page.

Math Course Policies: It is your responsibility to be aware of Math Course policies, including information on Text Policy, Attendance Policy, Disability accommodations, Make-up policy, Grading policy, the Honor’s Policy, and the Academic Integrity Policy.

5. Conclusion and Final Remarks

The focus of this section is to introduce and discuss Banach and Hilbert spaces, which play a crucial role in various results utilized by physicists. We will also delve into the definition of Sobolev spaces and demonstrate their significance in addressing linear problems in partial differential equations. Throughout this discussion, we will provide concrete examples and to facilitate a comprehensive understanding of these concepts. While the utilization of generalized functional analysis will be sporadic, we will ensure a detailed introduction to related concepts for clarity and coherence with the overall essay.