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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1answer
32 views

Show that $v_i^2 > v_i v_{i+1}$ where $\vec{v} \in R^m$

I am trying to show that the matrix $$ B_h = \left[ \begin{array}{cccc} 2 & -1 & 0 & 0 \\ -1 & 2 & -1 & 0 \\ 0 & -1 & \ddots & \ddots \\ 0 & 0 & \ddots ...
1
vote
1answer
79 views

Accumulation point in $\mathbb R^k$

We know that $p\in X$ is an accumulation point of subset $E$ of metric space $(X,d)$ if every neighborhood of $p$, meet $E$ on uncountably many points. Now in $\mathbb R^k$ with standard metric it ...
3
votes
3answers
143 views

Non Uniformly Elliptic Equations page 117 [G-T]

Let $\Omega\subset\mathbb{R}^n$ be open and bounded. Suppose also that $\Omega$ satisfies the exterior sphere condition at $x_0$ and let $B=B_R(y)$ be a ball such that $B\cap\overline{\Omega}=x_0$. ...
4
votes
0answers
88 views

Open map in Banach algebra

I'm having trouble showing a certian function is open and can be extended. Let $\Omega$ be a completely regular topological space and $A=C_b(\Omega)$ the space of all complex-values bounded ...
0
votes
1answer
47 views

Does the given $f_{n}$ converge uniformly

Let $f_{n}:\mathbb{R}^{2} \to\mathbb{R}$ be defined by $$f_{n}(x,y)=\frac{1}{n^{2}} \cdot \frac{x}{1+x+y}$$ Does $\sum_{n=1}^{\infty} f_{n}$ converge uniformly? For testing the converges of $\sum ...
2
votes
1answer
256 views

lsc function on compact set it attains its maximum minimum?

Is this true if so how to show it? if not true can you give a counter example: A lower semicontinuous function f on a compact set K attaings its minimum on K. A lower semicontinuous function f on a ...
2
votes
1answer
46 views

On the existence of a bounded linear functional

Let $\mathcal{H}$ be a Hilbert space. By the Riesz Representation Theorem, we have that any bounded linear $\psi \in \mathcal{H}^{*}$ is of the form $\psi(h) = \langle h, g \rangle$ for some $g \in ...
3
votes
2answers
118 views

Existence of a point such that $f(c) = \sqrt{\frac{1}{b-a} \int_{a}^{b}f^{2}(x)dx}$

This has been stumping my calculus class: If $f$ is continuous on $[a,b]$ and $f(x)\geq 0$ on $[a,b]$ show that there exists a $c$ in $[a,b]$ such that $f(c) = \sqrt{\frac{1}{b-a} ...
5
votes
4answers
806 views

Convergence of a Fourier series

Let $f$ be the $2\pi$ periodic function which is the even extension of $$x^{1/n}, 0 \le x \le \pi,$$ where $n \ge 2$. I am looking for a general theorem that implies that the Fourier series of $f$ ...
0
votes
2answers
68 views

Existence of a certain bouned linear functional in the dual of a Hilbert space

For any vector $h \in H$, where $H$ is a Hilbert space, show that $\exists$ a bounded linear functional $\psi \in H^{*}$ such that: $$\|\psi\| = 1 \ \text{and} \ \psi(h) = \|h\|$$ Can anyone ...
0
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1answer
341 views

Topological degree

I need help for this exercice 1)Let $\Omega$ be an open and bounded set from $\mathbb{R}^n$ and $f\in C(\overline{\Omega})$ ,we suppose that there exists $x_0 \in \Omega$ such that :if for $x\in ...
3
votes
0answers
57 views

Copies of finite sets in sets of positive measure

We say a set $A \subseteq \mathbb{R}^n$ contains the pattern of a finite set $B \subseteq \mathbb{R}^n$ if there exists a shift $t \in \mathbb{R}^n$ and scale $s > 0$ such that $t+sB \subseteq A$. ...
9
votes
5answers
599 views

the sum: $\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$ using Riemann Integral and other methods

I need to prove the following: $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots+(-1)^{n+1}\frac{1}{n}+\cdots=\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n}=\ln(2)$$ Method 1:) The series $\sum_{n=1}^\infty ...
2
votes
1answer
209 views

limit problem: $\lim_{n\to +\infty}n\int^1_0 x^nf(x)dx $ [duplicate]

Let $f:[0,1]\to\mathbb R$ be a continuous function.I want to calculate this limit:$$\lim_{n\to +\infty}n\int^1_0 x^nf(x)dx $$
2
votes
1answer
54 views

Separation of function

When can a function of 2 variables say $h(x,y)$ can be written as $$\sum_i f_i(x)g_i(y)$$ I want to know what conditions on $h$ would ensure this kind of separation.
0
votes
1answer
44 views

What is the inverse of the function $g(t)=1/(t+i)$

Let $t\in C$ and consider the map $$g(t)=\frac1 {t+i}$$ Find $g^{-1}$, the domain of $ g $ and the domain of $g^{-1}$? For $g^{-1}(x)$ I get $\frac 1 x -i$, but i think that is incorrect.
1
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2answers
64 views

$\lim\limits_{r\rightarrow \infty , 0} \left(\frac{\sum_1^n i^r/ n}{\sum_1^{n+1} i^r/(n+1)}\right)^{1/r} $?

Edited please help me! how can I evaluate: $$\lim_{r\rightarrow 0} \left(\frac{\sum_1^n i^r/ n}{\sum_1^{n+1} i^r/(n+1)}\right)^{1/r}$$ and $$\lim_{r\rightarrow \infty} \left(\frac{\sum_1^n i^r/ ...
2
votes
1answer
53 views

Is this estimation correct?

I have to estimate the following quantity $$\frac{|e^{i\sqrt{\lambda+i\varepsilon}|x|}-e^{i\sqrt{\lambda}|x|}|^2}{|x|^2}$$ in $\mathbb{R}^3$ ($\lambda>0$) where ...
1
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0answers
76 views

Attempt at Proving A Lemma (critical point, 2nd derivative, global maximum).

could you please check my attempt at proving the following lemma. Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ has two continuous derivatives, has only one critical point $x_{0}$ and ...
2
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1answer
112 views

Result relating to Stirling's Formula

Have a question that I'm stuck on here. Let $$r_n= \frac{\sqrt{n}}{n!}\left(\frac{n}{e}\right)^n$$ Express $\log\left(r_{n+1}/r_n\right)$ as simply as possible. For this I got ...
1
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1answer
95 views

Convergence sup norm on $[0,1]$

How to show the following: When $f_{n}$ converges to $f$ uniformly in $[0,1]$ where $f_{n}$ and $f$ are continuously differentiable THEN derivative of $f_{n}$ converges to derivative of $f$ ...
1
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2answers
30 views

Proving that $x\mapsto \sum_{y\in A\cap\left(0,x\right]}r\left(y\right)$ is right-continuous

Let $r:A\rightarrow\left(0,\infty\right)$ be defined on a countable infinite subset $A\subseteq\mathbb{R}$. Let $T:\left(0,\infty\right)\rightarrow\mathbb{R}, \ T(x)=\sum_{y\in ...
2
votes
1answer
84 views

A statement equivalent to the definition of limits at infinity?

I was fiddling around with the definition of limits at infinity and believe I have found a statement that is equivalent to the definition. So the question is this: are the following two statements ...
0
votes
2answers
67 views

Existence of the limit from the left of real distribution functions

Let $f$ be a distribution function (i.e. non-decreasing an right-continuous) on the real line. In this question, for example, it is proved, that the set $D$ of points of discontinuity of $f$ is then ...
1
vote
1answer
87 views

Behavior of the pointwise norm of the gradient w.r.t. to boundary conditions in elliptic PDEs

Let $B\subset \mathbb{R}^2$ be some open ball in the interior of a (nice) domain $\Omega$ and $y_i\in H_0^1(\Omega)\cap H^2(\Omega)$ for $i=1,2$. If I know that \begin{align} &\bullet\quad ...
2
votes
3answers
175 views

Convergence of improper integral involving exponential

How to show that $\int_0^{\infty} e^{\lambda x}x^{r} dx$ converges when $\lambda$ is negative and $r$ a positive integer?
0
votes
1answer
34 views

If a function restricted to every set of an open cover is $C^k$, then is the function $C^k$

I know that a function $f:X\longrightarrow Y$ is continuous if $f|_{U_{\alpha}}:U_\alpha\longrightarrow Y$ is continuous for every $U_\alpha$, where the collection $\{U_\alpha\}$ is an open cover for ...
3
votes
1answer
76 views

Find adjoint operator

Let $\mathcal{H}$ be a separable Hilbert space with orthonormal basis $(e_n)$. Denote by $V$ the subspace of finite linear combinations of the basis-vectors. Define $T$ on $\mathcal{H}$ with $D(T) = ...
1
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2answers
356 views

$\limsup c_n\le \max(\limsup a_n,\limsup b_n)$

have a question that im stuck on here Let $a_n, b_n$ and $c_n$ be three sequences of real numbers. Suppose $k_n \in [0,1]$ for all $n$. Let $$c_n = (k_n)(a_n) + (1-k_n)b_n\;.$$ Assuming that ...
1
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2answers
332 views

Uniform convergence of integral on unbounded interval

Suppose I have $\;f_n:[a,\infty)\to\mathbb{R}$ and $\int_a^\infty f_n(x) dx$ exists. If $f_n\to f$ uniformly on $[a,\infty)$, I am able to show that $\int_a^\infty f(x)dx$ exists and $\int_a^\infty ...
0
votes
1answer
31 views

Are there smooth functions with countable supports?

Does there exist a smooth differentiable function $f: \mathbb R \rightarrow \mathbb R$ whose support $cl \{x \in \mathbb R: f(x) \neq 0 \}$ is a countable set?
4
votes
3answers
277 views

A bounded sequence whose sequence of averages does not converge

Can we find a bounded sequence $\{a_n\}$ such that the sequence of its averages, say, sequence $\{b_n\}$, where $$b_n=\frac{1}{n}\sum_{i=1}^n a_i,$$ does not converge?
1
vote
1answer
63 views

Complex math-curves

Let $y$ be the curve in $X$ defined by $\displaystyle y(m)=\left(\frac{m^3}{3}\right)-m+\left(im^2\right)$, $m \in [-2,2]$ a)Let $\,f(z) = -iz\,$. Compute integral $\,(y) f(z)dz\,$ b)Calculate the ...
1
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3answers
2k views

Is the function $f(x)=\sin(1/x)$ differentiable at $x=0$?

The function $f$ is defined by $f(x)= \sin(1/x)$ for any $x\neq 0$. For $x=0$, $f(x)=0$. Determine if the function is differentiable at $x=0$. I know that it isn't differentiable at that ...
2
votes
1answer
55 views

Is reflection or rotation in a $2$-dimensional normed space isometric?

Is reflection in the $x$-axis or in the line $y=x$ in a $2$-dimensional normed space isometric? How about rotation through a right angle? If so, what is the proof?
3
votes
2answers
165 views

Uniform Convergence of integrals

If a sequence of functions $f_n$ are uniformly convergent in a given interval $[a,b]$ to a function $f$, are all riemann integrable, then the integral $$\int ^b_af_ndx\rightarrow\int^b_afdx$$ and $f$ ...
4
votes
5answers
81 views

Power series infinity at every point of boundary

Is there an example of a power series $f(z)=\sum_{k=0}^\infty a_kz^k$ with radius of convergence $0<R<\infty$ so that $\sum_{k=0}^\infty a_kw^k=\infty$ for all $w$ with $|w|=R$ Thank you ...
2
votes
1answer
41 views

Specific question about the consequence of composing power series

Please bear with my possible abuse of notation/terminology. Consider the power-series composition f(g(x)). If g's range lies within f's interval of convergence, and if series g has a constant term 0, ...
1
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3answers
247 views

Answer Check: Third degree taylor polynomial at the point (1,0,-1)

I found the third degree taylor polynomial $$f(x,y,z) = xy^2z^3$$ at $$(1,0,-1)$$ The answer i got $$p_3(x,y,z)=(\frac{1}{2!}(-2y^2))+ (\frac{1}{3!}(-2(x-1)y^2))+(\frac{1}{3!}(6(z+1)y^2))$$ I am not ...
0
votes
1answer
185 views

Using the general binomial theorem to find a series-like expression for $\sqrt 2$

How do I use the general binomial theorem (i.e. the series expansion of ${(1+x)^\alpha}$ for $ |x|<1$) to show the following? $$\sqrt 2=1+\frac 1{2^2}+\frac{1\cdot3}{2!\cdot{2^4}} ...
1
vote
1answer
155 views

Find the value $\sum_{m=1}^{\infty}{\frac{e^{-a m^2}}{m^2}}$

How to find the following series' value? $$\sum_{m=1}^{\infty}{\frac{e^{-a m^2}}{m^2}}$$ I know that this series converges. I check it by ratio test or comparison test for $$ a \in \left [ 10^{-15} ...
0
votes
1answer
27 views

Paramertrization of intersection between spehere and plane.

I have the normal $n = (a,b,c)$ for a plane through origo,and want to find the paramertrization of the unit circle. How can I do this? I guess I should eliminate one coordinate from the plane and ...
2
votes
1answer
71 views

Completing a proof

Say we are given this: Impossibility of ordering the complex numbers. As yet we have not defined a relation of the form $x < y$ if $x$ and $y$ are arbitrary complex numbers, for the reason that it ...
6
votes
2answers
138 views

If $f:\mathbb{R}\to\mathbb{R}$ is continuous and $[a,b]\subset f([c,d])$, how to prove there is some $[r,s]$ such that $f([r,s])=[a,b]$?

Let $f:\mathbb R\to\mathbb R$ satisfy the following: $f$ is continuous there exist closed intervals $[a,b]$ and $[c,d]$ such that $[a,b]\subset f([c,d])$ How to prove that there ...
2
votes
3answers
354 views

Prove integral is greater than $0$

$f(x)$ is Riemann integrable on $I=[a,b]$ and $f(x)>0$ for all $x \in I$, prove $\int_a^b f(x) dx >0$ . Need help on this question, please help me
2
votes
0answers
34 views

bonus question on quaternions [duplicate]

I have this asterisk question, I know its hard to do and I know no one would get it in my class. Just wondering if any of you guys could give me good hints in how to do this. It would be appreciated. ...
4
votes
4answers
1k views

How to show the intersection of arbitrary compact sets is compact in a general metric space?

I understand that if you are working in $\mathbb{R}^n$, then the intersection of an arbitrary collection of compact sets is compact because it is closed and bounded. But what if you are not in ...
2
votes
1answer
213 views

Finding Extra Condition for a function to satisfy $f(n)=n$

Given a function $f$ defined on the set of all natural numbers $\mathbb{N}$ with three conditions: If $m,n$ relatively prime, then $f(mn) = f(m)f(n)$. $f$ strictly increasing. $f(2) = 2$. Find a ...
2
votes
2answers
123 views

Orthogonal Polynomials

Let $\{P_n\}_{n=0}^{\infty}$ a family of polynomials in $[a,b]$ where $n$ is the degree of $P_n$. Suppose they are orthogonal with respect to a positive wight function $\rho$. Show that $P_n$ has $n$ ...
0
votes
0answers
55 views

Limit points of a sequence contained in $S^1$

Let $\theta\in (0,2\pi)$ be a real number such that $\displaystyle\frac{\theta}{\pi}\notin\mathbb{Q}$. We define $z:=\cos(\theta)+i\sin(\theta)\in S^1\subseteq\mathbb{C}$ and let ...