5
votes
0answers
99 views

Is there a book only about epsilon delta proofs?

I want to know if there is such book, with beautiful epsilon delta proofs of all kind.
1
vote
0answers
29 views

Extending a measurable map $f: G \to \mathbb{R}/\mathbb{Z}$ to a continuous group homomorphism

Let $G$ be a compact abelian metrizable group (where the group operation is written as $+$) and $\mu$ is the Haar measure on $G$. Suppose we have a measurable function $f: G \rightarrow ...
0
votes
0answers
25 views

Examples of elements in the Dirichlet space

By Dirichlet space we mean $\{F\in C_{0}[0,1]:\text{ there exists }f\in L^{2}[0,1]\text{ with }F(t)=\int_0^t f(x) \, dx$, $\forall t\in [0,1]\}$. The more the better. Any famous examples? Using the ...
4
votes
1answer
82 views

Proof for and Intuition behind Taylor's Theorem

I notice that multiple versions of a theorem are called Taylor (univariate/multivariate, approximate/exact). But I do not find it trivial to infer proof of one version from the rest. So looking for a ...
1
vote
1answer
95 views

Measure Theory Book

What book should I use for measure theory?I have solved Rudin's Principle Of mathematical analysis up to chapter 7.Some people advised me to use Real and complex analysis by Rudin, while other said it ...
2
votes
0answers
53 views

Interesting examples of switching limit and integral

We learn many theorems regarding the relationship of limit and integral (Dominated/ Monotone Convergence, Fatou, Semicontinuity of norms, etc...). As I'm working on my research, I find that I often ...
5
votes
0answers
39 views

Reference for the fact that a smooth function analytic on every line is itself analytic

Let $f \in \mathcal C^\infty(\mathbb R^p)$ ($p \geq 2$) be a smooth function such that the functions $g_d(t) := f(td)$ are all analytic for all $t \in \mathbb R$ and all $d \in \mathbb R^p.$ (i.e. $f$ ...
3
votes
1answer
62 views

Integral of products of cosines

Given $m+1$ integers $\alpha_0,\ldots,\alpha_m\geq 1$, I was trying to get a nice closed formula for the integral $$ \int_0^\pi\cos(\alpha_1\theta)\cdots\cos(\alpha_m\theta)d\theta. $$ More precisely, ...
1
vote
1answer
51 views

Existence of periodic orthogonal basis in $L^2([0,1])$ which is not trigonometric?

Let $$ \psi(x) := \sin(\pi x). $$ It is well-known that system $\{ \psi(n x) \}_{n \in \mathbb{N}}$ forms an orthogonal basis in $L^2([0,1])$. My question is the following: Are there other examples ...
5
votes
1answer
67 views

If $f: A\to\mathbb R$ one-to-one but not monotone, there exist $x,y,z\in A$ with $x<y<z$ such that $f(x) < f(y)$ and $f(y) > f(z)$ (wlog)

The following result is part of the folklore, but I'd like to have a standard reference for something that I am writing: If $A \subseteq \mathbb R$ and $f: A \to \mathbb R$ is one-to-one but not ...
1
vote
1answer
123 views

How do you self-study Functional Analysis?

It would be very handy to know about function spaces, distributions and Fourier analysis. It looks like Rudin's Functional Analysis covers these things, but I do not yet have the foundation for it. ...
0
votes
2answers
141 views

Where can I find SOLUTIONS to real analysis problems? [closed]

I'm specifically interested in problem sets in Real Analysis that have solutions. I have a few books on it, but I'd like to compare my solutions with some given answers in a lot of cases to ensure ...
0
votes
1answer
28 views

Isoperimetric inequality with Green-capacitiy

I was wondering what the progress is, in isoperimetric inequalities for Capacities, specifically with the Green kernel ( optional: and Riesz kernel with $a\in (2,\infty)$). Or if it is solved already, ...
1
vote
2answers
27 views

Text on convergence theorems in probability theory (various modes of convergence)

I need a text reviewing theorems and discussing with details ALL the types of convergence in probability theory such as almost sure convergence, convergence in probability, weak convergence, $L^p$ ...
2
votes
3answers
91 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
5
votes
1answer
65 views

Limit everywhere, limit function is continuous, specific proof.

Suppose $f:[a,b] \to R$ is a function such that $\lim_{t\to x} f(t) = g(x)$ exists $\forall x \in [a,b]$. It can be shown that $g(x)$ is a continuous function. I seem to remember that there was a ...
1
vote
1answer
43 views

Where can I find introductory video lectures about calculus and analysis?

I am having calculus classes that are titled as Calculus for Mathematicians, for the rest of the students who are studying calculus, they use Stewart's book. In our classes, we're having something ...
1
vote
3answers
98 views

Suggestions for a real analysis reference.

Can anyone suggest some real analysis book which has a geometric presentation of the concepts with pictorial representation.
0
votes
0answers
71 views

Multivariable calculus and real analysis in one semester. What is the best way to study for such course?

I am in my first year of EECS and planning on taking a lot of maths classes. I have already taken single variable calculus and linear algebra and did well in them and decided to take multivariable ...
0
votes
0answers
26 views

Modifying a Density Function

Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria: $$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad ...
3
votes
1answer
60 views

Is there a garden of derivatives?

I've found a book called A Garden of Integrals, in which the author shows the evolution of the concept of Integral. I follow AnalysisFact on Twitter, some days ago, they posted the following: The ...
0
votes
0answers
19 views

Nature of Hessian of a function of a matrix

If input to a differentiable function is a matrix, what is the nature of Hessian of the function? Is it a tensor or something? This is a simple question, but I guess I am not sure where refer to, to ...
1
vote
1answer
41 views

Looking for “explicit” integrals solvable using lebesgue integration theory

I am preparing for an exam in Measure and Integration Theory (Lebesgue Integration). As far as I know my professor prefers to ask students solving explicit integrals which can be solved using the main ...
0
votes
1answer
37 views

Introduction to bump functions/ mollifiers

I want to introduce a small perturbation into a particular point of a smooth function, and then, use bumo functions to smooth out the perturbed function. Could any of you recommend a good ...
1
vote
2answers
52 views

Book on periodic Schrödinger operators

I am looking for good books about the spectral theory of periodic (1-dimensional) Schrödinger operators on a compact interval. A good reference I found was Reed/Simon Analysis of Operators (and a ...
2
votes
1answer
40 views

Is there a general algorithm to solve computable integral equation?

Hilbert's tenth problem ask for the general algorithm(finite number of operation) to solve of all Diophantine problems.Today, it is known that no such algorithm exists in the general case. What ...
14
votes
2answers
327 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
4
votes
0answers
33 views

Weak topology on $L^p,~p> 1$

How looks like the weak topology in the particular case $X=L^p$, I mean, is possible to detail this topology beyond standar form: Arbitrary union of finite intersections open pre-images of opens ...
1
vote
1answer
121 views

Real analysis text books

I'm sure his has been asked before but can someone recommend a real analysis text book with lots of worked examples practice questions and solutions? The sort of textbooks I've used and really ...
0
votes
0answers
19 views

reference for existence and blow up results in transport-like PDEs (Vlasov equation)

I'm looking for references to results regarding maximal time existence of solutions of a certain transport-like PDE, more precisely this one: $$ \partial_t f + v \cdot \nabla_x f + E(f,t,x) \cdot ...
4
votes
4answers
139 views

A question regarding Frobenious method in ODE

Suppose $b(x),c(x)$ are real functions analytic at 0. Let $b(x)=\sum_{i=0}^\infty b_ix^i, c(x)=\sum_{i=0}^\infty c_ix^i$ on $(-R,R)$. Suppose $r$ is a double root of $r(r-1)+b_0r+c_0=0$. It is well ...
1
vote
1answer
33 views

Kolmogorov's Existence Theorem

My analysis professor told us to take the following theorem for granted in order to prove other results, but I would like to see a proof of it, since I think it will be beneficial. Here is the ...
1
vote
1answer
76 views

Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...
0
votes
1answer
29 views

Zeros of polynomials and power series

Consider $k_i \in \{-1,1\}$ for every $i \in \mathbb{N}$ and consider the family of polynomials $P$ of the form $$\mathop{\sum}\limits_{i=0}^n k_it^i;$$ and the family of power series $S$ of the form ...
5
votes
1answer
73 views

eulers original derivation for the Euler–Maclaurin formula?

Please does someone know a good description of how Euler did derive his summation formula? Thank you!
1
vote
0answers
64 views

Cannot find a good intermediate real analysis textbook with enough examples/solutions - Any suggestion?

I am currently taking an intermediate real analysis course, one that is between an undergraduate real analysis course and a graduate real analysis course in terms of difficulty. However, I have not ...
3
votes
0answers
99 views

Which book is better, Rudin or Ahlfors? [closed]

As a beginner, with no history in analysis, what book is better for self teaching; Rudin or Ahlfors? Thanks!
1
vote
0answers
20 views

smoothnes of rotationally symmetric functions

Consider a smooth function $f : [0,\infty[ \to \mathbb R$. This function induces a rotationally symmetric function $F : \mathbb R^2 \to \mathbb R$ via $F(x,y) = f(||x,y||)$. It should be correct ...
7
votes
1answer
138 views

How to learn inequalities and become good at proving them?

I am taking a real analysis course next year and I want to start slowly preparing for that class now, so I hope you can help me. The class is quite challenging and the fail rate is relatively high. ...
0
votes
1answer
28 views

Analyticity of Logarithmic Integrals

Assume $f\in L^2[0,1]$ and let $g(x)=\int_0^1f(y)\ln|x-y|dy$. Is it true that $g\in C^\infty(0,1)$? Is it true that $g$ is analytic in $(0,1)$? Can you refer me to a right reference to look up such ...
0
votes
1answer
34 views

How do i prove “change of variables”?

To be honest, I don't get the proof in Folland, "Real Analysis", p.74. Let $\|\cdot\|$ be the max norm on $\mathbb{R}^n$. Let $\Omega$ be open in $\mathbb{R}^n$. Let $G:\Omega \rightarrow ...
5
votes
1answer
117 views

A bounded subset in $\mathbb R^2$ which is “nowhere convex”?

Let $F : \mathbb S^1 \to \mathbb R^2$ represents a simple closed curve $C$ in $\mathbb R^2$. The Jordan curve theorem says that the curves bounds a interior domain $\Omega$ and $\partial \Omega= C$. ...
6
votes
1answer
140 views

Is it possible to extend a $C^1$-function smoothly from any Lipschitz domain?

If $\Omega$ is a cube in $\mathbb{R}^n$ and $f\in C^1(\overline\Omega)$. By reflection one can extend such a function to all of $\mathbb{R}^n$ and the extenstion is in $C^1(\mathbb{R}^n)$. If ...
0
votes
0answers
34 views

Reference for power series

I would need some references for power series, Taylors series of elementary functions, derivation and integration of power series, convergence of sequences of functions and series of functions. The ...
0
votes
1answer
102 views

Reference request: Partition of unity…

I was looking for some material that could help me understand a real analysis course (1st year undergraduate). My teacher treated the following topics: Partition of unity Existence of regular ...
0
votes
2answers
47 views

Riemann Sphere/Surfaces Pre-Requisites

I have recently developed a large interest in everything to do with Riemann Sphere/Surfaces. I wish to understand the topic quite well but I know that I will need to read a good number of books on ...
0
votes
0answers
9 views

analytic property of periodic properties

Can any one suggest me some books in which I can see the analysis of periodic functions? I dont have any constrain in the domain or codomain. For example this book ...
0
votes
1answer
40 views

Generalization of the Riemann integral to functions on the sphere

I want to do something a bit strange: define the Riemann/Darboux integral for bounded real functions defined on the n-dimensional sphere. The catch: I cannot use Lebesgue integration theory to do ...
3
votes
2answers
205 views

Is there another Analysis book that is based on the Cartesian space $\Bbb R^p$

I am in the middle of a slightly ambitious attempt to learn Analysis on my own. I skimmed through Rudin(Baby), Chapman Pugh, William Wade, Stephen Abbott and Strichartz and ended up preferring the ...
6
votes
2answers
306 views

A proof of a theorem of Liouville

I need some reference for the proof of the following theorem attributed to Liouville: Theorem: Let $f(x):\Omega\longrightarrow \mathbb R^n$ a $C^2$ function where $\Omega$ is an open subset of ...