Comments on current projects are welcome

Chapters 12 and 13 are now citable.

Oct 21, 2017

I have added in Chapter 12 the final version of Change of Variable Theorem for finite-dimensional Euclidean spaces. I decided against including surface measure in Chapter 12, since I don’t have a general theory. I have proofread Chapter 13, and I added a result that the Fourier series is complete in L2((-π,π],IK). I think that I will leave Chapters 12 and 13 as is now, and I may add to these chapters in the future if I think of something interesting. One topic I definitely want to add is Fourier transform. Another would be Laplace transform. But they may not be fit in Chapter 13. But, my current top priority is Chapter 14, Probability Theory. I want to lay the foundation for my teaching and research in stochastic systems.

The Complete Current Notes Including Fundamental Theorems of Calculus

Oct 9, 2017

Here is the complete current notes. Everything up to Section 12.7 are correct, which you can cite. I did have some shuffling of propositions around due to new insight gained in the study. I am thinking about whether to impose the condition that an integral is said to exist if it is absolutely integrable. This is in line with the definition of σ-finite banach space valued measures. If the total variation is unbounded, then the measure is not defined. In the current version, an integral may exist even though it is not absolutely integrable. But, then, none of the usual operations that we took for granted for an integral is guaranteed to be valid. So, I am thinking about this change, which is a major rework of Chapters 11 and 12.

The Complete Current Notes

Jul 14, 2017

I just uploaded the entire notes that I have for the Real Analysis Reading Note project. Chapters 12 through 15 are not completed yet, and contains mistakes. Please don’t refer to those results. In Chapter 9, I recently noticed that my existing version on the web and on my desk has the wrong definition of C∞ at x0. When I work on papers and the book, I always have the correct notion of this definition in my mind. I don’t know how come I got the wrong definition in the print. I have corrected this definition on pp. 264 and any related statements that refer to this definition. Luckily, this mistake only affects Chapter 9. So, corrections are not that much. I really suspect that it was done by some wannabe slave owner here in my city. But, I did remember that I thought about this definition when I did the notes. I am not sure. Sometimes I wonder why some fellow researchers showed disgust toward my mathematics skill. The above might be the reason. They read my notes and found mistake(s) like this one! This is why I want to leave Cincinnati. This place is evil.

A section on analytic functions in Chapter 9.

Oct 6, 2015

I have added a section on analytic functions at the end of Chapter 9. In this section, we properly give the definition of analytic function on normed linear spaces and then show that the composition of analytic functions is again an analytic function. The definition easily guarantees that the elementary functions in normed linear spaces are analytic. We then continue on to prove the analytic versions of Inverse Function Theorem and Implicit Function Theorem.

Chapter 11 is now complete (1)

Apr 19, 2013

I have updated Chapter 11 yet again for the following purposes. Propositions 11.140 and 11.190 and Theorem 11.196 are added for sharper results on the normed linear space of finite Y-valued measures on a measurable space. Propositions 11.186 and 11.187 are added for the generation process of σ-finite topological measure space from sequence of finite topological measure spaces. Proposition 11.185 is new. Propositions 11.137, 11.138, 11.145, and 11.146 are added to introduce the notion of vector measure. I am working on Chapter 12 and will include the new chapter after I complete an iteration on that chapter.