Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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48 views

Determine the area of $\phi (A) $

Let $ A=[0, 1] \times[0, 1] $. Let $ h $ be a continous function on $\mathbb{R}$ and let $\phi $ be defined by $$\phi (x, y)=(x+h (x+y), y-h (x+y))$$ Determine the area of $\phi (A)$.
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58 views

Roots of a polynomial plus a logistic equation

I would like to know if there are any methods to find the roots (analytically) of complex valued equations of the following form: $$ f(z)=P(z)+\frac{e^{-z}}{(1+e^{-z})^2} $$ where $P(z)$ is a ...
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2answers
60 views

Convergence of polynomials

Consider $x\in [0,1]$. I want to know if, in the limit as $n\rightarrow\infty$, the following remains a polynomial: $$\sum_{k=1}^{n}\frac{x^k}{k!}.$$ For any finite n, this is a polynomial. But will ...
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1answer
78 views

Why are rational singletons nowhere dense on the real line?

I'm trying to understand this using the definition that the interior of the closure must be non-empty for a nowhere dense set. Thanks
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2answers
373 views

When is the infimum of the sum of two sets equal to the sum of their infima?

When is the following true? $A$ and $B$ are subsets of real numbers. I don't say that $A$ and/or $B$ are closed: $$\inf (A + B) = \inf (A) + \inf(B)$$ When is there a strict inequality in between? ...
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1answer
102 views

How to find extrema points

I took notes in my class on finding extrema points, but I don't understand what I wrote in my notebook. I need to learn their solution methods before my exam tomorrow (this is not homework or ...
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3answers
104 views

A non-constant, increasing function $f$ such that $f(b)=\int_a^bf$

Is there a non-constant, increasing function $f\colon A\to B$, where $A,B\subset\mathbf{R}$ such that $$f(b)=\int_a^bf(x)\;\mathrm{d}x$$ for $a,b\in{A}$ with $a<b$.
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1answer
100 views

An estimate for $\ln(1+f(x))$ using Taylor expansion

A crucial skill for every aspiring analyst (like myself) is confidence in estimation - knowing when, where, and how to use tools like Big-and-little-O to gain quick upper bounds. I'm trying to push ...
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1answer
95 views

Sublinear functional as supremum of linear functionals

Given a sublinear functional on a Vector space $V$, is it possible to write it as supremum of family of linear functionals?
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1answer
126 views

Continuous functions as regulated functions: a property.

In Differential and Integral by Paul Lorenzen (1971) pag. 148, I read ... every continuous function is trivially approximable by step functions that have no jump at a given arbitrary point .... All ...
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1answer
77 views

A question about the concept of tangent plane from William Wade's book

This question is from William Wade's book 11.6.9 page: 435. I have the book's solution manual. That's, I have the question's answer. But, the answer is complicated accourding to me. I dont ...
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2answers
141 views

Conversion of sum of series into product form

Show that the following series and product are equivalent: $$ \sum_{n=1}^\infty \left[ \dfrac{1}{n(n+1)} \right] = \dfrac{1}{2} \prod_{n=2}^\infty \left[ 1+\dfrac{1}{n^2-1} \right] $$ Thought of ...
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1answer
44 views

Showing the existance of the functions $u,v,s,t$ by Implicit function theorem.

I am studying from William Wade's introduction to analysis book the question 11.6.5 at page 434 Question: The given nonzero numbers $x_0, y_0, u_0, v_0, s_0, t_0$ which is simultanuously equations ...
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1answer
547 views

Proof for Strong Induction Principle

I am currently studying analysis and I came across the following exercise. Proposotion 2.2.14 Let $m_0$ be a natural number and let $P(m)$ be a property pertaining to an arbitrary natural ...
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0answers
60 views

If someone asked, and if I do t understand its soution, then, how do i understand? Do I not have aright to ask again? [duplicate]

First of all, I searched the question, and someone asked, I found its solution. But I think, that solution is not clear enough. Forvexample, there, the reason why the integral is zero is not ...
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2answers
82 views

What is the definition of the norm

Let $x$ and $y$ be in $ \mathbb{R}^{n}$. I know from the definition of norm that $\|x\|=\sqrt{\sum_{1}^{n}x_{i}^{2}}$. Can anyone tell me what will be the norm of $\|x-y\|$? Is it ...
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0answers
64 views

Determine whether this class of holomorphic functions is starlike

According to Singh and Singh [1], the class of holomorphic functions $f$ in the unit disk such that $f(0)=f'(0)-1=0$ and $\mathrm{Re}\{f'(z)+zf''(z)\}>0$ is a subclass of starlike function. My ...
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2answers
288 views

Definition of accumulation point

I have here a definition of accumulation point: A point $x$ in a metric space $M$ is called an accumulation point of $A \subset M$ if every neighbourhood of $x$ contains some point of $A$ distinct ...
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1answer
198 views

The Denjoy Theorem

I'm currently studying Denjoy's theorem, which says the following: Theorem If $f$ is a diffeomorphism of $S^1$ with a irrational number rotation $ \rho$ and the variation of $f^{'}$ (denoted by ...
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1answer
63 views

General solutions of the equation $ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0$

What is a general solution of the equation $$ f(x+h,y+h)-f(x+h,y)-f(x,y+h)+f(x,y)=0 \textrm{ for } x,y \in \mathbb R, h>0, $$ with unknown function $f: \mathbb R^2 \rightarrow \mathbb R$? ...
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3answers
92 views

What are the differences between the Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$ and the standard Lebesgue measure on $[0,1]^n$?

There are no Lebesgue measure on infinite dimensional Banach space. However, there is a Lebesgue measure on the Hilbert cube $[0,1]^\mathbb{N}$. What are the differences between this measure and the ...
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1answer
86 views

a simple analysis question [closed]

This is the problem. Is this true? If it is true, prove it. If $ f $is differentiable at $ [0,1] $ , and $ f(0)=0$ then $ \int _0 ^1 f(x)^2 dx \le {1 \over 2} \int _0 ^1 f'(x) ^2 dx $ .
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121 views

Has this Principal Component Analysis (PCA) been done correctly?

I have a set of 3D data points, indicated by the blue color in the picture below. I then project them onto the x-y plane, i.e. setting z values of all the points to 0, shown by the yellow color ...
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1answer
77 views

Why base analysis on the natural numbers rather than on sets?

I'm looking Bishop's Foundations of Constructive Analysis. I decided to look at the Amazon's page searching for some elucidative review about if it's a worthy read, I've found this review. The first ...
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1answer
97 views

Show if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is linear and invertible

I want to show that if $\|\cdot\|$ is a norm then $\|f(\cdot)\|$ is a norm where $f$ is a linear and invertible function. First I need to show if $x\neq0$ then $\|f(x)\|>0$. Since $f$ is ...
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1answer
91 views

Showing a norm preserving isomorphism of vector spaces

Lets define $l^1$ as the complex vector space of all absolutely summable sequences of complex numbers and and $x_0$ consists of all the sequences in $l^\infty$ (all bounded sequences) that eventually ...
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1answer
156 views

Showing triangle inequality for a norm

I want to determine whether the following is a norm or not: \begin{equation} ...
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2answers
88 views

How to define an interior point in terms of $\epsilon$-balls?

Which is the technically correct definition? I) An interior point of a set $B$ is a point that is the centre of some $\epsilon$-ball in $B$. II) An interior point of a set $B$ is a point that is in ...
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1answer
147 views

Prove that $\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$

Let $f$ be a continuously differentiable function on $[0,1]$ and $f(0)=0$. Prove that $$\int_{0}^{1}|f(x)f'(x)|dx\leq\frac{1}{2}\int_{0}^{1}|f'(x)|^2dx$$ Thank you!
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274 views

Precise definition of epsilon-ball

My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$." Is this correct? ...
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1answer
60 views

Numerical integration over an actual series, is this possible?

I was wondering about the following: Is there any clever way to evaluate integrals over a series? Let me choose an example: $$\int_{\frac{2\pi}{3}}^{\pi} \sum_{n=0}^{\infty} P_n(\cos(x))dx$$ where ...
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1answer
70 views

Does a neighbourhood need to be a *connected* set?

I have in my topology/ real analysis textbook the definition of neighbourhood of a point as an open set containing that point. But isn't a neighbourhood necessarily a connected set? Wikipedia also ...
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144 views

Relation between the exponential function and the modified bessel function of second kind

I found the following sentence at the wikipedia page : Unlike the ordinary Bessel functions, which are oscillating as functions of a real argument, Iα and Kα(this is the mod. bessel function of the ...
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1answer
64 views

$|A(n)|<B$, $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ imply $\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$

Suppose that $|A(n)|<B$ and $\lim_{s \to 0^{+}}\sum_{n=1}^{\infty}a_{n}n^{-s}=a$ where $A(x)=\sum_{n \leq x}a_{n}$. Then $$\lim_{x \to \infty}\sum_{n \leq x} a_{n}(1-\log{n}/\log{x})=a$$ What I ...
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50 views

Fundamental Optimization question consisting of two parts.

A) Find all extrema of $$f(x)=\sum_{k=1}^{n} x_{k}^{2} $$ subject to the constraint $\sum_{k=1}^{n}\vert x_k\vert^p=1$ B) prove that $$\frac{1}{n^{(2-p)/(2p)}}(\sum \vert x_k\vert^p)^{(1/p)}\le (\sum ...
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1answer
82 views

Constructing a vector field

I'm doing a list of exercises, but there is one problem I can't solve. Let $$\Omega=\Big\{(x,y,z)\in\mathbb{R}^3 \ | \ \frac{x^2}{4}+\frac{y^2}{9}+\frac{z^2}{16}<1\Big\}.$$ Suppose that ...
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1answer
72 views

I want to calculate the integral of following function;

I WANT TO calculate the integral of following function: $$\begin{cases} 1 & \text{if } (x,y)\in I \text{ and } y > x, \\ 0 & \text{if } (x,y)\in I \text{ and } y\le x \end{cases}$$ By ...
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1answer
56 views

For $(X, d)$ and $f \colon X \to \mathbb R$, $\sup\,\{f(x) \mid x \in X\}-\inf\,\{f(x) \mid x \in X\}=\sup\,\{|f(x)-f(y)| \mid x, y \in X\}$

I would like to show that for a metric space $(X, d)$ and a function $f \colon X \to \mathbb R$, $\sup\,\{f(x) \mid x \in X\}-\inf\,\{f(x) \mid x \in X\}=\sup\,\{|f(x)-f(y)| \mid x, y \in X\}$. So ...
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1answer
71 views

About the existence of a certain type of linear map

I have a conjecture inspired by the following observation. If $f:\mathbb{R}\to \mathbb{R}$ is a continuous bijective function that satisfies $f(x)+f^{-1}(x)=2x$ and has a fixed point, then $f(x)=x$. ...
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139 views

Prove that $\int_{I}f=0 \iff$ the function $f\colon I\to \Bbb R$ is identically $0$.

Let $I$ be a generalized rectangle in $\Bbb R^n$ Suppose that the function $f\colon I\to \Bbb R$ is continuous. Assume that $f(x)\ge 0$, $\forall x \in I$ Prove that $\int_{I}f=0 \iff$ the function ...
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33 views

estimates for sums

How can I prove the estimate $ \int\limits_0^{2\pi } \vert \sum\limits_{j=0}^k \ r^je^{ij \theta} \vert \,d\theta \leq \int\limits_0^{2\pi } {\vert \sum\limits_{j=0}^k \ e^{ij \theta} \vert ...
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4answers
318 views

Show that $f$ is integrable and $\int_{I} f=0$

Let $I$ b a generalized rectangle in $\Bbb R^n$ Suppose the bounded function $f:I\to \Bbb R$ assumes the value $0$ except at a single point $x \in I$ Show that $f$ is integrable and $\int_{I} f=0$ ...
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1answer
43 views

Evulating $\int_I f$ by using Darboux Sum convergence Criterion

I tried to solve the question. But, there may be some mistakes. I want to learn this properly. If there exist any notation mistake, typo, a general mistake in solution way or else, please correct ...
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2answers
92 views

Use the Integrability Criterion to show that the function $f: I \to \Bbb R$ is integrable.

Question: For the generalized rectangle $I= [0,1]\times [0,1]$ in the plane $\Bbb R^2$ $$f(x,y)=\begin{cases} 5 & if\ \ (x,y)\ is\ in\ I\ and\ x> 1/2 \\ 1 & if\ (x,y)\ is\ in\ I\ \ and\ ...
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74 views

Averages of a Function

Let $(X,M,\mu)$ a measure space and $f:X \rightarrow \mathbb{C}$ a function in $L^{\infty}(\mu)$. Define $A_f$ as the set of all averages \begin{equation} \frac{1}{\mu(E)} \int_{E} f d \mu ...
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138 views

How prove this Mathematical Analysis by Zorich, from the chapter on continuous functions.

Let $P_n$ be a polynomial of degree $n$. For a function $f:[a,b]\to\mathbb{R}$, Let $\Delta(P_n) = \sup_{x\in[a,b]} |f(x)-P_n(x)|$. and $E_n(f) = \inf_{P_n} \Delta(P_n)$. A polynomial $P_n$ is the ...
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3answers
1k views

Proving Every open set in $\Bbb R$ is a countable union of open intervals. [duplicate]

This question is from William R. Wade's Introduction to Analysis book: Prove that every open set in $\Bbb R$ is a countable union of open intervals. I have no ideas honestly. Thank you.
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0answers
38 views

sequence of smooth functions bounded in $L^p$

I have the following situation : Consider $E \subset R^n$ a compact set. Let $\Omega_1 \supset \Omega_2 \supset ... \supset \cap_i \Omega_i = E$ a sequence of bounded open sets Supoose too ...
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1answer
110 views

The alternate triangle inequality in normed and metric spaces?

In real and complex analysis, we used the alternate triangle inequality $\left|a-b\right| \geq \left||a|-|b|\right|$ a fair bit. I've just been introduced to the notion of real inner product spaces, ...
4
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2answers
89 views

Prove that f is constant on $K$ that is, if $a \in K$ then $f(x)=f(a) \ \ \forall x\in K$

Suppose that $f: \Bbb R^n \to \Bbb R^m$ and that $a\in K$, where $K$ is a compact connected subset of $\Bbb R^n$ suppose for each $x\in$ $K$, $\exists$ $\delta_x >0$ such that $f(x)=f(y)$ ...