Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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81 views

Strictly positive solution of linear equations

Suppose $A\in\mathbb{R}^{m\times n}$, $b\in\mathbb{R}^m$, and $b\in \mathcal{R}(A)$. Show that there exists an $x$ satisfying $x \succcurlyeq 0$, $Ax = b$ if and only if there exists no ...
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1answer
39 views

Stronger condition then ultrametric condition on metric space

A metric space $(X,d)$ is called an ultrametric space if it is a metric space and fulfills the stronger triangle inequality (see Wikipedia) $$ d(x,y) \le \max\{ d(x,z), d(z, y) \}. $$ Examples are ...
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2answers
169 views

Example of continuous function that isn't uniformly continuous and isn't 1/x

I understand that in an open interval the only functions that are continuous but not uniformly are functions whose limits are singularities. But when we have a function $f:H\rightarrow\mathbb{R}$ and ...
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1answer
292 views

Clopen sets in $\mathbb{R}^n$

Are there any further clopen sets in $\mathbb{R}^n$ besides $\mathbb{R}^n$ and the empty set? $\mathbb{R}^n$ shall carry the topology that is induced by the canonical metric. So far, I could not find ...
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0answers
71 views

Equivalent definitions of Fourier transform of a measure

For me the fourier transform of a measure $\mu\in\mathcal{S}'(\mathbb{R})$ is defined by $\hat{\mu}(\varphi)=\mu(\hat{\varphi})$ where $\varphi\in\mathcal{S}(\mathbb{R})$. My question is: if one has ...
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3answers
53 views

How to prove that $\lim_{x\to 0}f(x)=L$ is equivalent to $\lim_{x\to 0}f(x^3)=L$ rigorously [closed]

I've tried to prove this by epsilon-delta, but it didn't go well...
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6answers
2k views

Every compact metric space is complete

I need to prove that every compact metric space is complete. I think I need to use the following two facts: A set $K$ is compact if and only if every collection $\mathcal{F}$ of closed subsets with ...
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2answers
120 views

Is this set open or closed or neither of both in $\Bbb{R}$?

I'm solving this problem in Rudin's Principles of Mathematical Analysis: and came up with a idea that is $\{p \in \Bbb{Q} \mid 2 < p^2 < 3 \}$ open or closed in $\Bbb{R}$ or neither of both? ...
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1answer
114 views

Find $\sum_{k=1}^{\infty}\frac{1}{z_k^2}$

Let $z_1, z_2,\dots, z_k,\dots$ be all the roots of $e^z=z$. Let $C_N$ be the square in the plane centered at the origin with siden parallel to the axis and each of length $2\pi N$. Assume that ...
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5answers
340 views

Puzzled by $\displaystyle \lim_{x \to - \infty} \sqrt{x^2+x}-x$

I am preparing for the next Semester and therefore review a few of my Analysis I limits, I have found this example in C.T. Michaels Analysis I: Compute $ \displaystyle \lim_{x \to - \infty} ...
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1answer
141 views

left sided limit of a monotonic function

Definition: Let $X,Y\subset\mathbb R$ and $f \colon X\to Y$. Let $x$ be a limit point of $X\cap(-\infty,x)$. If for all sequences $x_n\in X\cap(-\infty,x)$ it holds $f(x_n)\to y$, then $\lim_{x\to ...
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1answer
68 views

norm of Frechet derivative in point.

Let $ E = \mathcal B([0,1], \Bbb R) $ with supremum norm. Now I can define function $ F:E \ni f \rightarrow ||f||^2 - f(0) \in \Bbb R$ My task: 1)Show the differentiability of $F$ in: $ f_0: ...
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3answers
219 views

Limit of $\displaystyle \frac{1-\sin x}{1+\sin x}$

I have to find the limit of $\displaystyle \frac{1-\sin x}{1+\sin x}$ as $x\to\infty$. I was studying this function finding its real graph but to do this I need to know where the function goes as $x$ ...
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1answer
65 views

Zeta function in complex analysis.

Show that $$\frac{\zeta'(z)}{\zeta(z)}=-\sum_{n=2}^{\infty}\frac{f(z)}{n^z}$$ for $\Re z\gt 1$ Where $f(z)= \ln p$ if $n=p^m$ for some prime $p$ and some $m\in \Bbb N^+$ Or $f(z)=0$ otherwise. ...
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1answer
162 views

Raabe's test, logarithm test, Bertrand test

Raabe's test, logarithm test and Bertrand test are the most commonly used criterion in calculus. The relationship between them is quite interesting. Here is how: $\sum\limits_{n=1}^\infty a_n$, ...
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1answer
37 views

differences between Banach spaces and $\Bbb R^n$.

Can you please tell me, what are the biggest differences between Banach spaces and $\Bbb R ^n$? I am trying to understand the Frechet derivative.
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1answer
58 views

spectral structure of sinusoidal model

let us consider following code ...
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2answers
81 views

Bounded function on $(0,1)$ but discontinuous at $0$

Problem: Let$\ $ $f:(0,1)\rightarrow \mathbb{R}$$\ $ be bounded but such that $\lim_{x\rightarrow 0}f(x)$ does not exist. I need to show that there are two sequences ($x_{n}$) and ($y_{n}$) such that ...
2
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1answer
95 views

Subspace of Tempered Distributions

Let ${S_{h}}'(\mathbb{R}^{n})$ be the space of tempered distributions such that if $u\in {S_{h}}'(\mathbb{R}^{n})$, then $\lim_{\lambda\rightarrow \infty}{||\phi(\lambda D)u||_{\infty}} = 0$ for all ...
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0answers
71 views

Some standard elliptic estimates

Consider the problem $$\Delta_{g}u - c(n)R_{g}u + Ku^{p} = 0 \mbox{ at } M$$ where $c(n) = \frac{n-2}{4(n-1)}$, $K$ is a constant, $R_{g}$ is a scalar curvature, and $(M^{n},g)$ is a smooth ...
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1answer
118 views

A question about Heine-Borel Theorem.

Im reading chapter2 of rudin's Principle of Mathematical analysis. Heine-Borel theorem is involved in this chapter, $\mathbf{2.41}\,\,$ Theorem$\,\,\,$ If a set $E$ in $R^k$ has one of the ...
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1answer
148 views

“Proof” that $g(t) = (t,t,t,…)$ is not continuous with uniform topology

Let $g : \mathbb R \to \mathbb R^{\omega}$ be the function $$ g(t) := (t, t, t, \ldots). $$ If $\mathbb R^{\omega}$ is equipped with the uniform topology, and $\mathbb R$ with the standard topology, ...
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0answers
106 views

Are any of those quotient rings isomorphic to other well known rings?

(1) Let $C_b(\mathbb{R})$ be the ring (with pointwise multiplication and addition) of bounded continuous functions. Let $I_0=\{f_{(x)} \in C_b(\mathbb{R}) \space | \space lim_{x \to \pm ...
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1answer
273 views

Example of a false proof when a Fourier series is not unique?

I am attempting to come up with an example to illustrate why one should care that a function has a unique Fourier series expansion. Inspired by the fact that one can rearrange terms in a ...
2
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1answer
88 views

Series $\sum_{n=1}^{+ \infty}\frac{z^{n}}{n}e^{n^{2}z}$

Let $$f(z) = \sum_{n=1}^{+ \infty}\frac{z^{n}}{n}e^{n^{2}z} \ \ \ \ ,z\in \mathbb{C}$$ I want to find the maximal region in which $f$ is holomorphic. I have a problem with the convergence in $\{+i, -i ...
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2answers
586 views

Why modern mathematics prefer $\sigma$-algebra to $\sigma$-ring in measure theory?

Actually, i posted the exact same question before (about a year ago), but now i lost my past account so i couldn't find the past post.. So i googled this, but i couldn't find a satisfying post. I'll ...
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3answers
67 views

Is the $\omega$-product of the set of irrationals compact?

We know that any product of compact spaces is compact. But, I wonder that the countable product of $\mathbb{P}$ can be compact since $\mathbb{P}$ is not compact?
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3answers
85 views

A problem about constructing $R$ from $Q$

Im reading Chapter1 of Rudin's Principles of Mathematical Analysis, 3rd ed and a little confusing on his construction of $R$ from $Q$, See the step3. "Define γ to be the union of all α∈A" means ...
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1answer
98 views

Sobolev spaces inclusion

I'm having trouble finding an answer related to Sobolev spaces that does not relate to duality. I'm looking for an answer to the following question: When (i.e. for what domains $\Omega$ or such) can ...
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2answers
709 views

Determine the convergence of $ \sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right] $

I'm having trouble determining the convergence of the series: $$ \sum_{n=1}^{\infty}\left[1-\cos\left(1 \over n\right)\right]. $$ I have tried the root test: ...
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1answer
44 views

Conversion between two types of line integrals

This is probably a silly (and maybe duplicate) question, but I haven't been able to find a standard answer so far. Let $f$ be a real-valued function on $\mathbb{R}^n$, and let $F$ be a function from ...
3
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3answers
310 views

Is the sphere compact?

Riesz' lemma gives us that in infinite-dimensional spaces no ball is compact. but what is about the sphere$=\{x \in X; ||x||=1\}$? can we say something about the compactness of the sphere in ...
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2answers
96 views

Region of convergence for the series of functions

Find out the region of convergence for the following series of function. $$\sum_{m=1}^\infty x^{\log (m)}$$ Here $x \in \mathbb{R}$. This is a series of function. I was trying to find out the radius ...
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3answers
170 views

What's wrong with this demonstration? (1 = -1) [duplicate]

What's wrong with this demonstration?: $$A \iff 1 = 1^1$$ $$A \implies 1 = 1^\frac{2}{2}$$ $$A \implies 1 = (1^2)^\frac{1}{2}$$ $$A \implies 1 = ((-1)^2)^\frac{1}{2}$$ $$A \implies 1 = ...
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1answer
99 views

Inequalities of the quantile function [closed]

I'm trying to rigorously prove the following inequalities involving the quantile function $Y(a) = \inf \{x \in\mathbb{R} : a \leq F(x)\}$ where $F$ is the distribution function: 1) $F(x) < a \iff ...
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1answer
39 views

How do I show that $f:\Bbb R \to \Bbb R^2$ given by $f(x)=(x^3, |x^3|)$ is of class $C^2$?

Hobbyist, working my way through Munkres's Analysis on Manifolds. In one example (p. 196), author claims that $f:\Bbb R\to \Bbb R^2$ given by $$f(x)=(x^3, |x^3|)$$ is of class $C^2$.
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1answer
58 views

How do you modify the value of the function on set of zero measure ( $[0,\frac 12]$) such that it become continuous function on $[0,1]$?

$$f(x) = \begin{cases} 1 & x \in [0,\frac 12] \\ 0 & x \in (\frac 12, 1] \end{cases}$$ The function is discontinuous at $\frac 12$. How do you modify the value of this function on set of zero ...
3
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1answer
57 views

Entire function missing 2 values

Suppose $f$ is entire of finite order, and doesn't assume $2$ values in the complex plane. Is there a way to prove that $f$ is costant without using Picard theorem ?
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1answer
94 views

The complement of a cartesian product and the product topology

I tried to prove that for topological spaces $X_{\alpha}, \alpha \in A$ for the product space $\prod_{\alpha} X_{\alpha}$ with the usual product topology if $A_{\alpha} \subseteq X_{\alpha}$ for all ...
0
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1answer
48 views

The limit implied by an inequality

If $1 \ge F(y) > 1 - \epsilon$ for each $\epsilon >0$, how does this prove that $\lim_{y\rightarrow\infty} F(y) = 1$ when F is a right-continuous non-decreasing function. Thanks
3
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1answer
93 views

Hyperplane which can separate two closed convex set

Define: $$E=:\ell^1(R)=\{x=(x_n)_n: \|x\|_1=\sum_{n=1}^{\infty}|x_n|<\infty\}$$ We have two closed convex sets $X,Y$ as subsets of normed vector space $\ell^1(R)$ with $X\cap Y=\emptyset$ and $Y$ ...
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1answer
62 views

Computing complex integrals

Can anyone help me out with this question: Let $f(z)=\frac{1}{1+z^2}$. Compute $\int_{\gamma_{r}} f(z)dz$ if $r>0$ and $\gamma_{r}$ is the border of {$z \in C: |z| \leq r, Imz\geq 0$}. We walk ...
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2answers
53 views

Lemma 2.5.5 Boas, Entire functions

I'm reading Boas, Entire functions, but I don't understand lemma 2.5.5, which states that $\sum_{1}^{+\infty}\frac{1}{r_{n}^{\alpha}}$ and the integral $\int_{0}^{+\infty}t^{-\alpha -1}n(t)dt$ ...
2
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1answer
100 views

Order of growth of $ \prod_{n=1}^{+\infty} (1-e^{-2\pi n}\cdot e^{2\pi i z})$

The order of an entire function $f$ id defined as $$ord ( f) = inf \{\lambda \geq 0 \ | \ \exists A, B > 0 \ s.t. \ |f(z)|\leq Ae^{B|z|^{\lambda}} \forall z \in \mathbb{C} \}$$ I have $F(z) = ...
2
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1answer
123 views

Relation between convex set and convex function

Let $E$ be an normed vector space and $A\subset E$ be a closed nonempty set. Define $$\phi(x)=\operatorname{dist}(x,A)=\inf_{a\in A}\|x-a\|$$ Prove that if $\phi$ is convex then $A$ is convex. ...
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1answer
222 views

lim sup of the square equals square of the lim sup

"Suppose that for the sequence of real numbers {an}, lim sup (an) = c > 0 Prove that lim sup (an^2) = c^2" For this question, I tried two ways: 1) Since c is the limsup of {an}, given e >0 and k>0, ...
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3answers
122 views

Topologies such that every point has a finite number of neighborhouds

Are there results known about topologies such that every point is just contained in a finite number of open sets (or neighborhoods)?
2
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1answer
73 views

Why is the derivative of an (everywhere differentiable) function on the real line the limit of a sequence of continuous functions?

If a function $g$ on $\mathbf{R}$ is everywhere differentiable, why is $f=g'$ the limit of a pointwise convergent sequence of continuous functions $f_n$? More generally, does this also hold for any ...
1
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1answer
51 views

A limit involving the exponent of convergence

Let $f$ be an entire non-constant function with at least one zero. If $\{z_{j}\}_{j\in \mathbb{N}}$ are the zeros of $f$, set $$b =\inf\left\{\lambda >0 \ | \sum_{j}\frac{1}{|z_{j}|^{\lambda}}< ...
1
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1answer
80 views

In the change-of-variables theorem, must $ϕ$ be globally injective?

In the above theorem, doesn't $\phi$ need to be injective too? The inverse function theorem merely implies that $\phi$ is locally injective -- is this sufficient? I ask because Marsden, in his ...