0
votes
0answers
48 views

Real Analysis and dynamics

I am looking for a textbook or similar resource that addresses the content in a rigorous graduate course in real analysis(at the level of Rudin/Royden) with the following criteria: No hand waving - ...
0
votes
1answer
27 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
28 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
97 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 $\partial \Omega= C$. ...
5
votes
1answer
104 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
30 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
92 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
43 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
26 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
190 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 ...
5
votes
2answers
269 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 ...
2
votes
4answers
384 views

Proof of Riemann-Lebesgue lemma

I read a book, and this mention to the following lemma of Rieman-Lebesgue type. Lemma. Let $-\infty<a<b<\infty$ and $f(x,y):[a,b]^2\to\mathbb R$ be an integrable and nonnegative function. ...
1
vote
1answer
21 views

Reference request to study Borel summation

Could someone recommend sources to learn about Borel summation procedure? Books, articles or reviews? I have a background in basic analysis.
3
votes
1answer
81 views

Exponentiation of Real Numbers?

I'm looking to learn Real Analysis on my own. Am reading Elements of Real Analysis by Bartle. I came across this project which defines the powers of real numbers i.e. exponentiation. Firstly I am ...
1
vote
5answers
150 views

What books on analysis after someone has finished all 3 by Rudin?

What books on analysis would people recommend after someone has finished all three by Rudin (Principles of Mathematical Analysis, Real and Complex Analysis, and Functional Analysis)? I am looking for ...
2
votes
3answers
70 views

Recommend me a text or webpage introducting gamma function throughly

Till now, i have learned abstract Integration, all basic properties of the (n-dimensional) Lebesgue(-Stieltjes) measure and the lebesgue integral is an extension of Riemann integral. Here's an ...
1
vote
0answers
143 views

Lecture Notes in Real Analysis

I understand that this question was partially addressed here but I would like to have a question dedicated to just real analysis. I am looking for both elementary real analysis (advanced calculus type ...
1
vote
0answers
76 views

Summary of divergent series summation methods and relations between them?

There are a number of methods of assigning sums to series that do not necessarily converge, e.g. Cesàro summation, Abel summation, Ramanujan summation, etc. (There is also the trivial method of only ...
1
vote
1answer
46 views

Proving that a function is a metric

Let $$p(x,y)= \left|\frac{1}{x} - \frac{1}{y}\right|$$ for $x,y > 0$. Prove that $p$ is a metric for $(0,\infty)$. This question is from Methods of Real Analysis, 2nd edition by Richard ...
1
vote
1answer
52 views

How should I prove $\operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr$ without using spherical coordinates?

Let $B_n:=\{x\in{\Bbb R}^n:|x|\leq 1\}$ and $S^n(r):=\{x\in{\Bbb R}^{n+1}:|x|=r\}$. Then we have the following formula $$ \operatorname{vol}_{n+1}B_{n+1}=\int_0^1 \operatorname{vol}_n S^n(r)dr. ...
0
votes
0answers
46 views

Uniform continuity of inverse in only one variable

Let $f:[0,1]\times[0,1]\to \mathbb{R}$ be a (uniformly) continuous functions. Denote the image of $f$ by $D_f:=\{(x,y): x\in[0,1] , 0\leq y \leq f(x,1)\}$ $f$ is such that the section $f_x$, i.e. the ...
1
vote
6answers
73 views

Noncircular construction of $e$ and $\ln$ for the real line

Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? For example, they can define $e=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition ...
0
votes
0answers
27 views

Reference for Power Series and Function Sequences Anaysis

I found the way of analysis from robert strichartz a very neat book, although I felt he rushed a bit on power series and sequences of function. I would like to know other recommendations for analysis, ...
0
votes
0answers
37 views

Calculus with leibniz notation

Is there a modern book covering in depth calculus and multivariable calculus (maybe also real analysis?) using only Leibniz ($dx$) notation?
0
votes
3answers
170 views

Exercise book for Elementary/Introduction to Real Analysis?

I'm currently doing a course in Elementary Analysis (Intro to real analysis). My course focuses on the topics: sequences, limits of functions, continuity, uniform continuity and ...
7
votes
1answer
172 views

Reference request: Behavior of power series at endpoints

I would like to find a calculus book (or a book on real analysis or advanced calculus) which has the following result: If a power series $\displaystyle\sum_{n=0}^{\infty}a_{n}x^{n}$ has a radius of ...
1
vote
0answers
24 views

Gaussian Smoothing Error and “Hard Analysis” Bounds

Let $p \in (0,\infty)$. Consider a function $f \in L^p([0,1])$, and let $$\phi_\epsilon(x) = \frac{\exp(-x^2/2\epsilon^2)}{\sqrt{2\pi\epsilon^2}}$$ denote a $0$-mean Gaussian of variance $\epsilon$ ...
3
votes
3answers
98 views

Cardinality of all sequences of non-negative integers with finite number of non-zero terms. (NBHM 2012)

Consider the set $S$ of all sequences of non-negative integers with finite number of non-zero terms. Is the set $S$ countable or not? What is the cardinality of the set $S$ if it is not countable? ...
0
votes
1answer
31 views

Referrences in $L^p$ spaces

I'm learning Real Analysis, and I want to practice some problems about $L^p$ spaces. Can you tell me some problem books or textbooks about it. I known some books such as Problems in Mathematical ...
7
votes
2answers
498 views

$L^1$ and $L^{\infty}$ are not reflexive

I want some proof for the following statement : $L^1$ and $L^{\infty}$ are not reflexive. Can anyone help me, please? or reference me?
4
votes
0answers
53 views

What is known about functions with alternating signs of partial deriavites?

Lately I have come accross a class of functions with alternating signs of their partial derivatives, in detail the class looks as follows $$\Big\{f:\mathbb{R}^n\longrightarrow\mathbb{R}: (-1)^k ...
4
votes
0answers
33 views

How much larger is the $\sigma$-algebra than the algebra in Caratheodory extension?

Given a 'measure' $\lambda$ on an algebra $\mathcal{A}$ of sets, Caratheodory gives a way to extend this $\lambda$ to a $\sigma$-algebra. The idea is we define an outer measure (on all subsets) ...
0
votes
1answer
80 views

Equality in Minkowski's theorem

I would like to see a proof of when equality holds in Minkowski's inequality. The proof is quite different for when $p=1$ and when $1<p<\infty$. Could someone provide a reference? Thanks!
1
vote
1answer
59 views

What is the best way to define the diameter of the empty subset of a metric space?

This question is related to Why are metric spaces non-empty? . I think that a metric space should allowed to be empty, and many authorities, including Rudin, agree with me. That way, any subset of a ...
0
votes
0answers
20 views

Lipschitz continuity of a minimum of a set of linear functions

I have a multidimensional function $A(h, u)$, that is linear in $u$ for any $h$, and convex in $h$. How can I prove that $\mathcal{A}(u) = \min_h A(h, u)$ is Lipschitz? If this is some standard ...
0
votes
0answers
28 views

Reference recommendation: Bernoulli polynomials and euler-maclaurin summation formula

I'm looking for a real analysis reference book that covers these areas. Any recommendations?
2
votes
1answer
56 views

Suppose that $0 \le f(n) \le 1$, why $\lim_{n \to \infty} (1 - f(n))^n = 0 \iff \lim_{n \to \infty} f(n)n = \infty$?

Suppose that $0 \le f(n) \le 1$ and consider two eqauation: $$ \lim_{n \to \infty} (1 - f(n))^n = 0 \tag{A} $$ $$ \lim_{n \to \infty} f(n)n = \infty \tag{B} $$ It seems that A and B are equivalent. ...
2
votes
3answers
181 views

Any suggestions about good Analysis Textbooks that cover the following topics?

I am an undergraduate math major student. I took two courses in Advanced Calculus (Real Analysis): one in Single variable Analysis, and the second in Multivariable Analysis. We basically used Rudin's ...
1
vote
3answers
205 views

From analysis of realvalued functions to analysis of Hilbert/Banach-valued functions

Does anybody know of a text (doesn't matter which form: article, book etc. - anything's welcome) in which it is described which result from real analysis also hold for Hilbert/Banach spaces ? I'm ...
1
vote
1answer
56 views

Where to find a proof of Silverman-Toeplitz?

I am referring to the theorem which gives a necessary and sufficient condition on a infinite matrix that maps convergent sequences to sequences converging to the same limits. Wiki gives a link to ...
0
votes
1answer
41 views

$C^1$-extension of function on a normal doamin

Let $f(x,t)$ be defined on the set $N:=\{(x,t): x\in(a,b), 0\leq t \leq g(x)\}$ where $g(x)\in C^1([a,b])$, $g>0$ and $f(\in C^1(\bar N))$. Is it possible to extend $f$ smoothly on the set ...
0
votes
0answers
45 views

Exercise references

I could recommend any good text analysis, or perhaps a list of exercises with good problems (for show) on dips, submersiones and implicit functions. I appreciate any references.
2
votes
1answer
65 views

Inverse problem in calculus of variations

I am interested in knowing which differential equations follow from a variational principle. I am reading this and it provides the answer for ordinary differential equations. Is there a complete ...
1
vote
4answers
131 views

real analysis rigourous definition of real numbers

I am enrolled in real analysis course in university. Our professor started right from the beginning defining the real numbers and all the usual operations on real numbers (like addition, ...
0
votes
0answers
101 views

What is the enough background in real analysis needed for this study in set theory?

In text , Classic Set Theory For Guided Independent study by Goldrei. It's said that the reader is expected to know real analysis. For those who know the text, what is the needed background in ...
3
votes
2answers
149 views

Good books to learn Riemann integration

I am looking for a good text book to learn Riemann integration. Please suggest books with theories and proofs comprehensively explained.
12
votes
2answers
274 views

Why don't fractals have more differentiable symmetries?

Some sets tend not to "look" very homogeneous, such as self-similar fractals. I'd like to know why! And there's a particular class of statements that I'm hoping can be made... Definition Let $A$ be ...
1
vote
1answer
48 views

Application of Stone-Weierstrass to approximate $f\in C(X\times Y,\mathbb{R})$ where $X$ and $Y$ are compact Hausdorff spaces?

On the wikipedia page for Stone-Weierstrass, the application section (http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Applications_2) says If $X$ and $Y$ are two compact Hausdorff ...
2
votes
0answers
46 views

Weakest Conditions for Convolution to be Differentiable

I was going through various posts about differrentiability of convolutions. What I would like to ask is: Suppose $f \in C^{1}(\mathbb{R})$. Then what conditions on the function $g$ would ensure that ...