Tagged Questions
1
vote
2answers
129 views
A less known definition of the definite integral of a continuous function
The definite integral of a continuous function can be defined using the bounded monotone sequence property: see Osgood's Functions of Real Variables, p.110.
(link to full book) (screenshots: page ...
0
votes
1answer
92 views
Which topics of real-analysis should be studied if you have already done calculus
Which parts of real-analysis are worth studying if you have already taken several calculus courses? I know that real-analysis is more 'rigurous', but still I wonder whether it is worth to again go ...
1
vote
1answer
19 views
Recurrence inequality for Dirichlet's eta function.
I'm studying the following function:
$\theta(p)=\eta(p)\eta(p-2)-\frac{p-1}{p}\eta^2(p-1)$,
where $\eta$ - Dirichlet's eta function. If we build a plot for $p\in [1,150]$, it's easy to see that it's ...
2
votes
1answer
41 views
Helpful to review certain calculus topics before first real analysis course?
This is my first time posting, so I apologize in advance if my question is inappropriate here. I wanted to know if it would be beneficial for me to review certain calculus topics before I take my ...
3
votes
2answers
76 views
What are the real-world applications of real analysis?
I've read the wikipedia article on mathematical analysis and this, but I can't exactly find an answer. Is real analysis just some pure math, or does it really have something to with physical ...
2
votes
1answer
56 views
Reference request: Newton-Kantorovich hypothesis for polynomials of integral coefficients
Kantorovich's theorem states that the Newton method for finding the
roots of a nonlinear function is guaranteed to converge if a
parameter $h$, determined by the values of the function and its
...
3
votes
1answer
40 views
normal form of an n-form
It is known, that one can convert any function $f(x_1,\dots,x_n)$, defined near $0$, into the function $(y_1,\dots,y_n)\mapsto a+y_1$, by a suitable local change of coordinates, provided $df\neq 0$. ...
3
votes
1answer
45 views
Elliptic regularity - nonlinear case
Let's say I have a weak solution $u \in H^1(\Omega)$, $\Omega \subset \mathbb{R}^n$ open, to the equation
$$
\Delta u = e^u,
$$
let's also assume that $e^u \in L^\infty(\Omega)$.
Does it follow that ...
4
votes
0answers
116 views
Self-study Real analysis Tao or Rudin?
The reference requests for analysis books have become so numerous as to blot out any usefulness they could conceivably have had. So here come another one.
Recently I've began to learn real analysis ...
0
votes
1answer
37 views
Similar textbook to Konigsberger's Analysis 2?
I am currently taking an introductory course to real analysis and my professor has decided to leave Rudin's "Principles of Mathematical Analysis" when teaching us the concepts of Lebesgue integration. ...
4
votes
1answer
81 views
Discreteness of eigenvalues for certain operators - can this approach be made rigorous?
I was idly thinking about why one might naïvely expect a discrete spectrum of eigenvalues for a linear operator $L$ when I dreamt up the following argument (which I expect isn't new instead - ...
1
vote
1answer
33 views
Continuous dependence of zeros on a parameter
Let $F:I\times J\to\mathbb{R}$ be a $C^k$ (or analytic) function, with $I,J$ real open intervals.
Set $f_\lambda(x):=F(\lambda,x)$ and consider the parametric equation
$$f_\lambda(x)=0\,.$$
Assume its ...
1
vote
4answers
106 views
Fourier Analysis
I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function?
Can someone provide me simple information ...
5
votes
0answers
97 views
Algorithm to calculate multiple integral.
One of the major difficulties of student in advanced calculus (including myself when student) is to obtain the extremes of repeated integrals to calculate the volume integral in $R^n$ i.e. transform ...
0
votes
1answer
21 views
Outer measure defined by a continuous and bijective function
This problem is from K.T. Smith's Primer of Modern Analysis:
Let $\psi: \mathbb{R}^d \to \mathbb{R}^d$ be continuous and one-to-one on an open set $\Omega \subset \mathbb{R}^d$ and define $$\nu(A) ...
3
votes
2answers
110 views
A sequel for Elementary Analysis by Ross?
I've been learning real analysis from this book:
Elementary Analysis, K.A. Ross
I really liked the style of this book. It is quite old, and sometimes very difficult, but I guess I liked the way it ...
0
votes
1answer
34 views
Sigma algebra of a regular borel measure
From the definition I am using and restrict to $\mathbb{R}^d$ only, what can we say about $\sigma$-algebra of $\nu$-measurable sets, $\mathfrak{B}_{\nu}$?
Some more specefic questions:
It contains ...
0
votes
0answers
32 views
References that discuss systems of ODEs on the non-negative orthant of $\mathbb{R}^n$?
Does anyone know of any references discussing initial value problems on the non-negative orthant? More specifically, consider the initial value problem
$\frac{dx}{dt}=f(x),\quad\quad ...
2
votes
1answer
56 views
Derivative of Regular Borel Measure
I just want to check if there are any other references on the definition of derivative of a regular borel measure beside the one I am reading:
A regular Borel measure on an open set $\Omega \in ...
0
votes
0answers
53 views
Simple optimization trick
Let $f,g:X\to\Bbb R$ be two functions where $X$ is any set. Then
$$
\left|\sup_x f(x) - \sup_x g(x)\right|\leq \sup_x|f(x) - g(x)|.
$$
This fact is fairly easy to prove, but it seems to be a ...
7
votes
3answers
200 views
Is $C([0,1])$ a “subset” of $L^\infty([0,1])$?
This is motivated from an exercise in real analysis:
Prove that $C([0,1])$ is not dense in $L^\infty([0,1])$.
My first question is how $C([0,1])$ is identified as a subset of $L^\infty([0,1])$? ...
8
votes
1answer
175 views
Real analysis textbok that develops the subject in a self-motivated, coherent fashion?
Well, it seems as though I just failed my analysis prelim for the second time... I have one more try in about $5$ months.
I'm failing to build up a framework for how to think about analysis problems. ...
2
votes
0answers
75 views
Books for Practice Problems
I'm not asking for the solution to the question posted below, but instead for references to books where I can find similar questions.
I'd be grateful for any advice you can give. Thank you.
3
votes
1answer
64 views
References on constrained least square problems?
I have met some constrained least square problems, for example, my last post. I found that there are various methods for slightly different constraints, and still I often had little clue about how to ...
1
vote
0answers
17 views
Almost Everywhere Convergence of Walsh Series of $L^2$ functions
I am currently reading the Hunt's papar (http://www.mathunion.org/ICM/ICM1970.2/Main/icm1970.2.0655.0662.ocr.pdf), and am wondering if there is some notes which presents his argument more ...
0
votes
0answers
31 views
Functional Analysis- Albert Wilanksy
Has anyone used this book? Can anyone recommend it for self-study? Thanks
9
votes
1answer
97 views
Is this (classical?) exercice missing a hypothesis?
A friend just told me about an exercice he was given quite a few years ago, but he wasn't sure wether he remembered all the hypothesis correctly. Does anybody recognize this?
Let $f$ be a smooth ...
10
votes
6answers
314 views
Video lectures on Real Analysis?
One of the most annoying gaps in my math education is Real analysis.
I tried hard, but all I could find are either Harvey Mudd College lectures or MathDoctorBob. The latter are too short and the ...
4
votes
1answer
47 views
Cylinders and dyadic Intervals
Let be $\Sigma_{+}=\{0,1\}^{\mathbb{N}}$ the space of sequences of $0^{'s}$ and $1^{'s}$. Consider the following surjective application
$\phi:\Sigma_{+}\to [0,1]$ given by
$$ ...
2
votes
0answers
50 views
Survey papers in real analysis
Can somebody recommend a good survey paper (or something similar to it) for real analysis?
I'm looking for something much shorter than a $200$ page text-book, but should still tell what the main ...
4
votes
1answer
191 views
A theorm about Cesàro mean, related to Stolz-Cesàro theorem
Original Title: Tauberian theorems and Cesàro sum
Theorem (Landau-Hardy, From Rudin's Principle of Mathematical Analysis Exercise 3.14)
$\newcommand\abs[1]{\left\lvert#1\right\rvert}$
If $\{s_n\}$ ...
3
votes
0answers
41 views
Where can I find the proofs of these measure theory Propositions?
These propositions are listed as examples in my script and I want to know the proofs (I am not able to prove them myself, unfortunately, because im not smart and very new to measure theory) :
If ...
3
votes
1answer
124 views
Question regarding the Kolmogorov-Riesz theorem on relatively compact subsets of $L^p(\Omega)$.
Usually, the Kolmogorov-Riesz theorem is quoted for $L^p(\mathbb R^n)$, but I am looking for versions considering spaces over subsets in $\mathbb R^n$.
The following is from the book "Sobolev spaces" ...
1
vote
2answers
78 views
Definition of $T_1$ Space in Kolmogorov-Fomin Introductory Real Analysis book
On page 85 of the book it reads the following definition:
Definition 4. Suppose that for each pair of distinct points $x$ and $y$ in a topological space $T$, there is a neighborhood $O_x$ of $x$ ...
0
votes
2answers
128 views
Approximating characteristic functions by continuous functions
The Urysohn Lemma is a very useful lemma,this lemma appears in several equivalent forms, one of them, what interests me is the following:
Uyshon Lemma: For every closed set $K$ in $X$ and every open ...
3
votes
2answers
209 views
Exercises on $\inf$, $\sup$, $\liminf$ and $\limsup$
I am studying analysis and having difficulties with $\inf$, $\sup$, $\liminf$ and $\limsup$ as you can see from the title. As discussed in a number of math.SE questions, definitions for sequences, ...
10
votes
4answers
287 views
Approximating continuous functions with polynomials
Given $\epsilon \gt 0$ and $f \in C^{0}[0,1]$, Weierstrass says that I can find at least one (how many? probably a lot?) polynomial $P$ which approximates f uniformly: $$\sup_{x \in [0,1]} |f(x) - ...
1
vote
1answer
241 views
Defintion of Upper & Lower Riemann Sum
I recently came across the terms: 'upper Riemann sum' and 'lower riemann sum'. Are they represent the same things as of 'upper sum' and 'lower sum' defined as follows.
4
votes
1answer
78 views
A reference for some fact in analysis
I am looking for a reference for the following fact. Any hints would be appreciated.
Suppose $(x_n), (y_n)\subset [0,1]$ are some sequences, $(a_n)$ is absolutely summable and for each $f\in C[0,1]$ ...
5
votes
2answers
147 views
Properties of certain type of measures
Let $p\in (0,1)$. For each $k\in\mathbb{N}$ and tuple $(\varepsilon_1,\ldots,\varepsilon_k)\in\{0,1\}^k$ denote
$$
S_{\varepsilon_1,\ldots,\varepsilon_k}=\left\{\sum\limits_{j=1}^\infty x_j 2^{-j}: ...
3
votes
1answer
174 views
choosing a real analysis text [closed]
I am trying to decided which real analysis text book I should buy.
I have the following in mind-
1)Walter Rudin's "Principles of Mathematical Analysis"
2)Charles Pugh's "Real Mathematical Analysis"
...
104
votes
3answers
5k views
The Integral that Stumped Feynman?
In "Surely You're Joking, Mr. Feynman!," Nobel-prize winning Physicist Richard Feynman said that he challenged his colleagues to give him an integral that they could evaluate with only complex methods ...
4
votes
2answers
225 views
Advice about taking mathematical analysis class
I appologize if this isn't the place to ask, if it's not could you let me know and I will take it to meta? Anyway, so I am planning on taking a mathematical analysis course next spring, and I'm really ...
2
votes
2answers
133 views
Can anybody recommend a website or other type of resource which contains real Analysis-type questions and their solutions?
In a recent post I asked about an epsilon delta proof for an Analysis question. Before posting it I searched for similar questions and their proofs on google but all I found were computational style ...
3
votes
2answers
136 views
Continuous functions on discrete product topology
Let $A = \{a_1,\dots,a_m\}$ be a finite set endowed with a discrete topology and let $X = A^{\Bbb N}$ be the product topological space. I wonder which bounded functions $f:X\to\Bbb R$ are continuous ...
2
votes
1answer
68 views
Finding a linear functional's corresponding measure
I just started looking through the proof of the Riesz Representation Theorem (in Rudin's Real and Complex Analysis), and I am still very confused about several things. I'll just write the statement of ...
1
vote
1answer
322 views
Principles of Mathematical Analysis
I just got through reading the first chapter of principles of mathematical analysis by Walter Rudin, the first chapter goes on and on about Dedekind cuts and then starts defining properties of them, ...
0
votes
3answers
121 views
References on relationships between different $L^p$ spaces
I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.
-1
votes
1answer
114 views
Infimum, supremum, limit
When I deal with sets of reals, i sometimes face with combination of $\lim \sup$ and $\sup$ and something like that.
Especially, $\lim \sup (\sup B_k) = \lim B_k$?
Would you recommend me where can I ...
1
vote
1answer
102 views
Discrete Sobolev Space and Sobolev Spaces of Banach Space valued functions
This is a reference request.
Can someone kindly give me some refernce(Books/papers) on
Discrete Sobolev Space (like we use Discrete $L^p$ spaces of $g\colon\Omega\to\Bbb R $ maps with norm given as ...
