Real analysis Definition and 509 Threads

We have ##R##, the set of Dedekind cuts which is an ordered set by proper inclusion. Fix an ##\alpha\in R## and let ##\beta## be the set of all ##p## with the following property: $$\text{there exists }r>0\text{ such that }-p-r\notin\alpha.$$ Rudin shows ##\beta\in R## and that...

I fail to see why the "by-product" of Proposition 4.6 is true. Isn't $$f_n\stackrel{L^p}{\to} f \text{ and } f_n(x)\to g(x) \ \mu(\mathrm{d}x)\text{-a.e.}$$ the same statement? I am confused about how to apply Proposition 4.6 to prove the "by-product" statement. Any help is greatly appreciated.

Let ##(f_n)## be a sequence of measurable functions from ##E## into ##\mathbb R##. I'm reading a proof of the fact that the set ##A## of all ##x\in E## for which ##f_n(x)## converges in ##\mathbb R## as ##n\to\infty## is measurable. The proof goes like this (I'm paraphrasing): Why is...

I'm reading a proof of a lemma that $$A_rf(x)=\frac1{m(B(r,x))}\int_{B(r,x)}f(y)\,dy,$$where ##m## is Lebesgue measure, is jointly continuous in ##r## and ##x## (##A## stands for average). The claim that ##\chi_{B(r,x)}\to\chi_{B(r_0,x_0)}## on ##\mathbb R^n\setminus S(r_0,x_0)## is made in the...

Here's an excerpt from the proof of the change of variables formula in Folland's book (Theorem 2.47, page 76, 2nd edition, 6th and later printings): For reference, see Theorem 2.40 below. I don't understand how he is using Theorem 2.40 in the quoted passage. Which part of Theorem 2.40 is he...

I am learning analysis from Rudin's famous book (baby rudin). I am confused about how ##\mathbb{R}## is defined in this book. In the appendix of chapter 1, he says that members of ##\mathbb{R}## will be certain subsets of ##\mathbb{Q}##, called cuts. Is this definition different from the way we...

Here's the inverse function theorem as stated in Spivak's book: Then there's a paragraph in Folland's book: I have read the inverse function theorem and its proof in Spivak's Calculus on Manifolds and I have a hard time reconciling it with what Folland states in his book on the chapter on...

I feel silly for asking, since I have accepted this always as true, but I don't have a reference for what ##0^p## equals when ##p## is a positive real number. This dawned on me when trying to show the positive definiteness of the ##p##-norm for ##x\in\mathbb R^n##, that is, $$x=0\iff...

Here's my definition I've been working on. Comments? Suggestions for improvements? EDIT: The reason I'm looking for a sequential characterization of right continuous is because the way you check that ##F## is right continuous is through...

I'm working the above exercise on the Lebesgue criterion for Riemann integrability in Folland's real analysis text, i.e. a function is Riemann integrable on ##[a,b]## iff it is continuous a.e. on ##[a,b]##. I think I know a solution, but I'm more concerned about elementary things. Are ##h,H##...

That's the first sentence in the proof. Prior to this Folland mentions the spaces ##L^1(\overline{\mu})## and ##L^1(\mu)## and how "we can (and shall) identify these spaces." (here ##\overline{\mu}## is the completion of ##\mu##). The propositions mentioned in the proof read as follows: I'm...

On page 45 in Folland's text on real analysis, he writes that we define Borel sets in ##\overline{\mathbb R}## by ##\mathcal B_{\overline{\mathbb R}}=\{E\subset \overline{\mathbb R}: E\cap\mathbb R\in \mathcal B_{\mathbb R}\}##. Then he remarks that this coincides with the usual definition of...

First some definitions: After Rudin has shown that ##\mathcal B## is an algebra, he invokes another theorem to claim it is uniformly closed. That theorem states that if ##X## is a metric space and ##E\subset X##, then ##\overline E## is closed. I don't understand Rudin's reasoning here, and...

I want to prove following (Big Picard Theorem forms):\ Theorem. The followings are equivalent:\ a) If ##f \in H(\mathbb{D}\setminus\{0\})## and ##f(\mathbb{D}') \subset \mathbb{C} \setminus \{0, 1\}##, then ##f## has a pole of an removable singularity at ##0##.\ b) Let ##\Omega \subset...

I need help please! So I'm reading 'Complex made simple' by David C. Ullrich. I made all the requirements for this proof but the author don't give the proof of this final theorem, instead it gives a similar proof for another set of theorems. Let ##\mathbb{D}' = \mathbb{D} \setminus \{0\}##...

##\textbf{Theorem}## If ##u: \mathbb{D'} = \mathbb{D} \setminus \{0\} \to \mathbb{R}## is harmonic and bounded, then ##u## extends to a function harmonic in ##\mathbb{D}##. In the next proof ##\Pi^+## is the upper half-plane. ##\textbf{Proof}##: Define a function ##U: \Pi^{+} \rightarrow...

I have a question about Daniel Fischer's answer here Why the function ##g(w)## is well-defined on ##\mathbb{D} \setminus \{0\}##? I don't understand how ##\log## function works here and how a branch of ##\log## function can be defined on whole ##\mathbb{D} \setminus \{0\}##. For example...

The below proposition is from David C. Ullrich's "Complex Made Simple" (pages 264-265) Proposition 14.5. Suppose ##D## is a bounded simply connected open set in the plane, and let ##\phi: D \rightarrow \mathbb{D}## be a conformal equivalence. (i) If ##\zeta## is a simple boundary point of...

I struggle with verifying positive-definiteness, in particular $$\langle f,f\rangle =0\implies f=0.$$ I know that for continuous non-negative functions, if the integral vanishes, then the function is identically ##0##. Here, however, ##f## being in ##L^2## does not make it continuous, right...

There are a lot of steps left out of this proof. Here is my proof with hopefully all the steps. I would like to know if it is correct Let ##A## be a countable set. Then ##A## is either finite or countably infinite. Case 1: ##A## is finite. There is a bijection ##f## from ##A## onto...

I am very unsure about the proof below. I'd like to know if it is correct. If ##B## is empty then it is finite by definition. If ##B## is non-empty then since ##B\subset\mathbb{N}## it has a smallest element ##b_1##. If ##B \backslash \{b_1\}## is non-empty then it has a smallest element...

These axioms lead to certain properties The properties above apply to all fields. We can define a more specific type of field, the ordered field And the following properties follow from this definition My question is about the proof of (c). My initial proof was Using b) with...

This problem is the final exercise of problem set 1 on MIT OCW's course 18.100A, Real Analysis. Since there are no solutions available for this problem set, I would like to show my attempt at a solution here and ask if it is correct. Here is the problem statement (also available as problem 6...

In Ordinary Differential Equations by Adkins and Davidson, in a chapter on the Laplace transform (specifically, in a section where they discuss the linear space ##\mathcal{E}_{q(s)}## of input functions that have Laplace transforms that can be expressed as proper rational functions with a fixed...

The above two video from numberphile are trying to motivate real analysis (I think?). The latter continues on from the former. Presenter's argument goes kind of like this: He first considers the real number line and talks about measuring distance between numbers 4-3=1 Then he talks about...

Hi everyone in the following expression ##f(t)=\frac{1}{2 \pi} \int\left(\int f(u) e^{-i \omega u} d u\right) e^{i \omega t} d \omega ## the book says I can't swap integrals bacause the function ##f(u) e^{i \omega(t-u)}## is not ## L^1(\mathbb{R} \times \mathbb{R})## why ? complex...

In these lecture notes, there is the following theorem and proof: I'm confused about "...the power series converges if ##0\leq r<1##, or ##|x-c|<R##...". In other words, why is ##|x-c|<R## equivalent to ##0\leq r<1##? I guess the author reasons as follows. If $$R=\lim _{n\to \infty...

I have to prove the associative law for addition ##(a+b) + c = a + (b+c)## using Peano postulates, given that ##a, b, c \in \mathbb{N}##. Now define the set $$ G = \{ z \in \mathbb{N} |\forall\; x, y \in \mathbb{N} \quad (x + y) + z = x + (y + z) \} $$ Obviously, ## G \subseteq \mathbb{N} ##...

Some time back I posted about my videos on Group Theory on YouTube and got valuable feedback from the PF community. With the response in mind, I made substantial changes to my presentation. One of the main complaints was that I was speaking too fast. Here is my recent video on Real Analysis...

When I learned calculus, the intuitive idea of infinitesimal was used. These are numbers so small that, for all practical purposes (say 1/trillion to the power of a trillion) can be taken as zero but are not. That way, when defining the derivative, you do not run into 0/0, but when required...

my question is how can I approch the problem ? And what is explicitly the set f(Eα)? {f(x) ∈ [a, b] such that what ??}

Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...

Proof: Suppose f is a function and x is in the domain of f s.t. there is a derivative at the point x and sppse. there are two tangent lines at the point (x,f(x)). Let t1 represent one of the tangent lines at (x,f(x)) and let t2 represent the other tangent line at (x,f(x)) s.t. the slopes of t1...

Hi. I'm nearing retirement so I thought I would take some math classes. This fall I took a Real Analysis class at a good school and dropped it because I did so bad on the first exam. I did great on the homework and quizzes. I also took Real Analysis about 47 years ago at a very good school...

Hello, PF! It’s been a while since I last posted. I am looking for a critique and recommendations regarding my study plan towards Functional Analysis and applications (convex optimization, optimal control), but first, some background: - This plan is in preparation for my master’s thesis, I...

Let ##S=\{s_n:n∈N\}##. ##\sup S## is the least upper bound of S. For any ϵ>0, we have an m such that ##\sup S−\epsilon \lt s_m## ##\sup S−s_m \lt \varepsilon## ##|\sup S−s_m| \lt \varepsilon## I mean to say that, no matter how small ϵ is, there is always an element of S whose distance from supS...

Suppose ##x \in \mathbb{Q}## and ##x > 1## and ## x^2 < 2##. I need to come up with some ##y \in \mathbb{Q}## such that ##x < y## and ## y^2 < 2##. Here is my attempt. Give that ##x > 1## and ## x^2 < 2##, I have ## (2-x^2) > 0## and ##4x > 0##. Also, ##2x >0##. Now define $$ \alpha = \text{...

## y-x \gt 1 \implies y \gt 1+x## Consider the set ##S## which is bounded by an integer ##m##, ## S= \{x+n : n\in N and x+n \lt m\}##. Let's say ##Max {S} = x+n_0##, then we have $$ x+n_0 \leq m \leq x+(n_0 +1)$$ We have, $$ x +n_0 \leq m \leq (x+1) +n_0 \lt y+ n_0 $$ Thus, ##x+n_0 \leq m \lt...

A Dedekind cut is a pair ##(A,B)##, where ##A## and ##B## are both subsets of rationals. This pair has to satisfy the following properties A is nonempty B is nonempty If ##a\in A## and ##c \lt a## then ##c \in A## If ##b \in B## and ## c\gt b## then ##c \in B## If ##b \not\in B## and ## a\lt...

Summary:: x Problem: Let ##f:[0, \infty) \rightarrow \mathbb{R}## be a positive function s.t. for all ## M > 0 ## it occurs that ## f ## is integrable on ## [0,M] ##. Which of the following statements are true? A. If ##\lim _{x \rightarrow+\infty} f(x)=0## then ##\int_{0}^{\infty} f(x) d x##...

Hi all. I am a math major. I will be taking real analysis next Fall with an excellent professor who I know to be also quite demanding. I would like to be as well prepared for this class as possible besides going through a real analysis text or lecture series over the Summer and causing the class...

I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ... I need some help with the proof of Proposition 6.1.2 ... and for this post I will focus on the first auxiliary result ... see (i) ... at the start of...

Problem: Let ## (X,d) ## be a metric space, denote as ## B(c,r) = \{ x \in X : d(c,x) < r \} ## the open ball at radius ## r>0 ## around ## c \in X ##, denote as ## \bar{B}(c, r) = \{ x \in X : d(c,x) \leq r \} ## the closed ball and for all ## A \subset X ## we'll denote as ## cl(A) ## the...

Prove that each of the limits exists or does not exist. 1. ##\text{lim}_{x\rightarrow 2}(x^2-1)=3## ##\text{lim}_{x\rightarrow 2}(x^2-1)=3## if ##\forall \epsilon>0, \exists \delta ## such that ##|x-2|<\delta \Rightarrow |f(x)-3|<\epsilon##. \begin{align}&|x^2-1|=|x+1||x-1|\leq \epsilon\\...

I'm learning Linear Algebra by self and I began with Apsotol's Calculus Vol 2. Things were going fine but in exercise 1.13 there appeared too many questions requiring a strong knowledge of Real Analysis. Here is one of it (question no. 14) Let ##V## be the set of all real functions ##f##...

I'm learning about Fourier theory from my lecture notes and I have a few questions that I wasn't able to concretely find answers to: 1. What's the definition of periodic extension? I think the definition is as follows ( Correct me if I'm wrong please ): for ## f: [ a,b) \to \mathbb{R} ## its...

Attempt: Note we must have that ## f>0 ## and ## g>0 ## from some place or ## f<0 ## and ## g<0 ## from some place or ## g ,f ## have the same sign in ## [ 1, +\infty) ##. Otherwise, we'd have that there are infinitely many ##x's ## where ##g,f ## differ and sign so we can chose a...

I have a sequence of functions ##0\leq f_1\leq f_2\leq ... \leq f_n \leq ...##, each one defined in ##\mathbb{R}^n## with values in ##\mathbb{R}##. I have also that ##f_n\uparrow f##. Every ##f_i## is the limit (almost everywhere) of "step" functions, that is a linear combination of rectangles...

Hi everyone, I recently started studying real analysis from baby Bruckner couple. It feels me like, "I am running too fast to reach my destination but in the process of running, I decreased my oxygen level." So, I stopped trying to complete uni coursework fast. But rather I started reading...

What are your opinions on Barry Simon's "A Comprehensive Course in Analysis" 5 volume set. I bought them with huge discount (paperback version). But I am not sure should I go through these books? I have 4 years and can spend 12 hours a week on them. Note- I am now studying real analysis from...