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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3
votes
2answers
405 views

Completeness of a normed vector space

This is captured from a chapter talking about completeness of metric space in Real Analysis, Carothers, 1ed. I have been confused by two questions: What does absolutely summable mean in metric ...
3
votes
1answer
106 views

basic exercise about Schwartz spaces

Let $f \in S(R)$ (the Schwartz space of rapidly decaying functions) such that $f(0)=0.$ Show that exists $g \in S(R) $such that $f(x) = xg(x)$. My try : By the calculus fundamental theorem $$ f(x) ...
2
votes
1answer
74 views

Limit of sums is sum of limits in a metric space

So I'm aware that in a normed space, the limit of the sums is the sum of the limits: For normed space $(X, ||.||)$, if $x_n \rightarrow a$ and $y_n \rightarrow b$, then $(x_n + y_n) \rightarrow ...
1
vote
0answers
32 views

Differential Equation Issue

Let there be $r(t)$ a differentiable function (over one variable), and let us define $f$: $$f(x,y) = x\cdot r\left(\frac{\partial y}{x}\right)$$ I need to find what expression this differential ...
2
votes
1answer
39 views

Quantifying Ill-posedness using Sobolev Space Estimates

I've been learning about ill-posed/inverse problems, and I'm having a hard time parsing/understanding the following, which seems crucial to the theory: Say we have an operator ...
1
vote
1answer
104 views

Let $f\colon E\to \mathbb{R}$ be continuous at $p\in E$. Prove that there exists a positive constant $M$

Let $f\colon E\to \mathbb{R}$ be continuous at $p \in E$. Prove that there exists a positive constant $M$ and $\delta > 0$ such that $|f(x)| \le M$ for all $x \in E \cap N_\delta(p)$. The book ...
0
votes
1answer
78 views

Let $E \subset R$ and let $f$ be a real-valued function on $E$ that is continuous at $p \in E$

Let $E \subset R$ and let $f$ be a real-valued function on $E$ that is continuous at $p$ in $E$. If $f(p) > 0$, prove that there exists an $\alpha > 0$ and a $\delta > 0$ such that $f(x) \ge ...
5
votes
1answer
75 views

Continuous complex functions.

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. Then $\partial(g(D))\subseteq g(\partial ...
1
vote
3answers
56 views

Is there a means of analytically proving the following identity?

Okay, so before I begin, my background is more in the world of applied, rather than pure, mathematics, so this question is motivated by a physics problem I'm looking at just now. Mathematically, it ...
0
votes
2answers
65 views

Limit of a recursively defined sequence

$ a_{n+1} = \frac{3+2a_n}{3+a_n} $ and $ a_0 = 1 $ This sequence is obviously increasing, so if we could prove it is bounded, we'd also prove it converges and we could easily find the limit by $ L = ...
0
votes
1answer
50 views

Determine the interpolating polynomial

Determine the polynomial of $ deg \le 6 $ interpolating function $$ f(x) = x^3 + 2x^2 + x + 1 $$ at the points : $ -3, -2, -1, 0, 1, 2, 3 $. My first idea it was to use Lagrange's formula, but it's ...
0
votes
1answer
48 views

Analysis-how to show that sequence is bounded

When we have the sequence $ a_{n}=kn-[mn]$, where $k$ and $m$ $ \in \mathbb{R} $. How could we show that the limit $ a_{n} $ exist? We have to show that the sequence is monotone and bounded,right? How ...
1
vote
1answer
200 views

Closure of equicontinuous family of bounded functions.

Let $B(x,y)$ be the set of all the bounded functions $f: X \to Y$ ($X,Y$ metric spaces). Prove that if $\mathcal F \subset B(x,y)$ is an equicontinuous family, then $\overline {\mathcal F}$ is ...
1
vote
1answer
64 views

Convex Analysis Question

I need to show that $F(x,y,z) = (y - z, z- x, x - y)$ is Lipschitz on the closed ball $S = \{(x,y,z): x^2 + y^2 + z^2 \le 1\}$. I get that F should be Lipschitz since S is closed and bounded (I ...
2
votes
1answer
76 views

Lipschitz function and uniform continuity on the real line

Let $\phi\colon\mathbb{R}\to\mathbb{R}$ be a function that satisfies $\lvert \phi(x)\rvert \geq 1$ for all real $x$ and $\lvert\phi(x) - \phi(y)\rvert \leq \lvert x - y\rvert $ for all real $x,y$ (for ...
2
votes
2answers
133 views

Show that $x \mapsto \frac{f(x)}{x}$ is strictly increasing on (0,1) given that f '(x) is strictly increasing on (0,1) and that f(0)=0

Let f : [0, 1] → R be a continuous function with f (0) = 0 and suppose that it is differentiable for all x ∈ (0, 1). Further assume that $x \mapsto f'(x)$ is a strictly increasing function on (0, 1). ...
1
vote
0answers
42 views

Imaginary and real part of some complex function

Let $\mu$ be some probability measure and $$g(t) = \log \int\exp(itx)\mu(dx) $$ How to see, that $\mathcal{Im}(g)$ is odd and that $\mathcal{Re}(g)\leq 0 \,\,\forall\ t\in\mathbb{R}$ Finally i ...
1
vote
1answer
100 views

prove that: $\lim_{x \to \infty} [f(x+1)-f(x)] = 0$ just by using definitions of limit and definition of derivative.

Let f and it is given that $\lim_{x\to \infty} f '(x) = 0$. I have to prove that: $\lim_{x\to \infty} [f(x+1)-f(x)] = 0 $ just by using definitions of limit and definition of derivative. I have no ...
0
votes
1answer
46 views

sequences-show that $\inf\{x_{n}: n\in \mathbf{ N}\}>0$

Knowing that $x_{n}>0$ for each $n \in\mathbf{N}$ and $x_{n}\rightarrow x>0$. Let $B$={$x_{n}:n \in\mathbf{N}$}. How could I show that $\inf B>0$??
2
votes
2answers
169 views

Calculus of variations, what is a functional

I'm reading a bit about the calculus of variations, and I've encountered this bit: Suppose the given function $F(.,.,.)$ is twice continuously differentiable with respect to all of its arguments. ...
3
votes
1answer
304 views

Find $I=\int_{0}^{2\pi}\frac{\cos^2\theta}{5+4\sin\theta}d\theta$

I've found by Cauchy's Integral Formula that $$\int_{\gamma (0,1)}\frac{\Re(z)}{2z-i}dz=-\frac{\pi}{4}i$$ but now not sure how to find $I$ using this? And help much appreciated!
1
vote
0answers
22 views

Some analysis concerning Levy-Kinchin-Formula (integral-identity)

Let $$\psi_n(t) = \int \left(\exp(itx)-1\right)\nu_n(dx)$$ be the characteristic function of a C-Poisson distribution. Defining $$ \psi(t) := \psi_n(t) -\frac{1}{2}\int_{t-1}^{t+1}\psi_n(s) \,ds $$ ...
2
votes
3answers
147 views

Find a sequence such that $\liminf a_n^{1/n}=1/4,\ \limsup a_n^{1/n}=1/3$

How to construct a sequence $\{a_n\}$ such that $1>a_n>a_{n+1}>0$ for all $n>0$ and $$ \liminf a_n^{1/n}=1/4,\ \limsup a_n^{1/n}=1/3. $$ Thanks.
0
votes
2answers
82 views

Calculate the limit of the function

Please help me to calculate the limit of the function. I do not know where to start. Thank you.
0
votes
2answers
108 views

A bounded sequence has a convergent subsequence

Let ${a_n}$ be a bounded sequence of real numbers. Prove that ${a_n}$ has a subsequence that converges to lim sup $a_n$. Thanks!
3
votes
1answer
129 views

Mean-value like theorem for holomorphic functions.

Let $f\in H(\Omega)$ for some open set $\Omega\subset\mathbb{C}$. Suppose $z\in\Omega$ and prove that there exists two distinct complex numbers $s,t\in\Omega$ such that ...
3
votes
3answers
132 views

Problem related to continuous complex mapping.

We are given with a map $g:\bar D\to \Bbb C $, which is continuous on $\bar D$ and analytic on $D$. Where $D$ is a bounded domain and $\bar D=D\cup\partial D$. 1) I want to show that: ...
2
votes
1answer
27 views

For $f\in H(U)$, find a bound on $|f(0)|$ given separate bounds of $|f|$ on $\partial U^{+}$ and $\partial U_{-}$.

Let $f$ be holomorphic in the unit disc $U$ and continuous on $\bar{U}$. Suppose $|f(e^{i\theta})|\leq1$ for $\theta\in[0,\pi)$ and $|f(e^{i\theta})|\leq\frac{1}{4}$ for $\theta\in[\pi,2\pi).$ Show ...
1
vote
1answer
2k views

Find the lim sup and lim inf of each of the following

just wanted to make sure I understood what I'm doing here. a. $5+(-1)^n$ Since -1 alternates between -1 and 1, then the smallest $a_n$ can be as $n->inf$ is 5-1 = 4, and the largest (sup) it ...
1
vote
2answers
115 views

Is a continuously differentiable function convex if all its partial second derivatives are non-negative?

I'm having trouble understanding the relevant Wikipedia article which begins with a convex set $X$ and then uses functions of single variables for succeeding examples; the MathWorld article seems to ...
0
votes
2answers
96 views

$a_j \geq 0, \sum_{j=1}^{\infty} a_j$ divergent $\implies \sum_{j=1}^{\infty}\frac{a_j}{1 + a_j}$

Suppose that $a_j \geq 0$ and that the $\sum_{j=1}^{\infty} a_j$ diverges. Prove that the following series diverges: $$\sum_{j=1}^{\infty}\frac{a_j}{1 + a_j}$$ Hint: first show that if it converges, ...
1
vote
3answers
89 views

Show and prove if the following series converges or diverges

Show and prove if the following series converges or diverges $$\sum_{j=1}^\infty{ \frac{(1+(1/j))^{2j}}{e^j}}$$ "I tried the comparison test, the root test, and the ratio test, but got messed ...
3
votes
4answers
799 views

Prove the set is not open.

Prove that the set $\mathbb{R}-\{1/n|n \in\mathbb{N}\}$ is not open. OK, so I am having a little trouble. I know that the definition of open set is : iff every point of $A$ is an interior point of ...
1
vote
1answer
73 views

Convergence of $|a_{n+1} - a_n| \le Cq^n$

Be $C\gt 0$, $0\le q\lt 1$ and $(a_n)_{n\ge 1}$ a sequence in $\mathbb R$ with $$|a_{n+1} - a_n| \le Cq^n$$ Show that $(a_n)_{n\ge 1}$ converges.
0
votes
2answers
62 views

simple undergraduate series quesiton

consider $ \displaystyle \sum_{n=1}^\infty (-1)^{n-1}a_n $ where $ (a_n) $ is a monotone decreasing sequence of nonnegative numbers with $ a_n \rightarrow 0 $ by the alternating series test, series ...
2
votes
0answers
23 views

Compute tetrahedral region

Show that the volume of region $A$ is $1/6$. Region $A$ is a tetrahedral region in $\mathbb R^3$. $$A=\{(x,y,z)∈R^3 \mid x\ge 0, y\ge 0, z\ge 0, \text{ and } x+y+z\le 1\}$$
0
votes
2answers
475 views

Why is every continuously differentiable function with a uniform bounded derivative lipschitz continuous

I only know how to prove this for functions on a convex set by using the mean value theorem, but is this also true for this general case when nothing is said about the domain of the function besides ...
-1
votes
1answer
52 views

$|a_n - a_m| \le \sum_{k=m}^{n-1} |a_{k+1} - a_k|$

How to show, that every for sequence $(a_n)_{n\ge_1}$ with $n \gt m$ following holds ? $$|a_n - a_m| \le \sum_{k=m}^{n-1} |a_{k+1} - a_k|$$
0
votes
1answer
152 views

Continuity of a constant function

If f assumes only finite many values, then f is continuous at a point $x_0$ if and only if f is constant on some interval $(x_0 - \delta, x_0 + \delta)$ I know how to prove continuity for a given ...
2
votes
1answer
114 views

Need a counter example for series convergence

I need some advice for constructing a counter example for $\sum\limits_{i=1}^\infty a_i$ converge but $\sum\limits_{i=1}^\infty \frac{a_i}{i}$ diverges.
0
votes
1answer
65 views

Counterexample to show that the change of variables for integral does not hold

Give a Counterexample to show that the change of variables formula does not hold if $g$ is not one-to-one even though $Jg(x) \neq 0$ where $Jg$ is Jacobian Matrix. [Hint: Take $f = 1$ and $g(x,y) = ...
0
votes
1answer
196 views

Quotient norm and actual norm

I have a question about the proof that $X\backslash U$ is a Banachspace if $X$ is one and $U$ is closed. In my book it is said, that for $x_k \in X$ and a series $\sum_{k=1}^{\infty}||[x_k]||< ...
2
votes
1answer
59 views

Functions, Continuity and IVT

Suppose that $g$ is a function defined and continuous on $\mathbb{R}$ and $n$ is a positive integer such that $$\lim_{x\to \infty} \dfrac{g(x)}{x^n} = 0 = \lim_{x\to -\infty} \dfrac{g(x)}{x^n}$$ (i) ...
0
votes
2answers
54 views

Functions and the IVT

Let $g, h$ be continuous functions defined on some interval $J$ and suppose that $g(x) \neq 0$ for any $x \in J$. If $g(x)^2 = h(x)^2$ for all $x \in J$, show that either $g(x) = h(x)$ for all $x \in ...
0
votes
0answers
29 views

estimate for a function in the H^s space

Consiser $f \in H^s(R^n)$ $(s> n/2)$. Show that : $$ |f(x) - f(y)| \leq \gamma_s(|x-y|) || f||_s , \ \ \forall x,y \in R^n .$$ Where $\gamma_s(|x-y|) = 2 (2 ...
2
votes
2answers
51 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
2
votes
1answer
167 views

Does the Weierstrass $\wp$ function have any double values besides $\infty$?

Given nonzero complex constants $\omega_1,\omega_2$, with nonreal ratio, we define $$\wp(z;\omega_1,\omega_2)=\frac{1}{z^2}+ \sum_\omega \frac{1}{(z-\omega)^2}-\frac{1}{\omega^2} $$ where the sum is ...
2
votes
3answers
391 views

What do physicists mean with this bra-ket notation?

In Quantum mechanics we said that $\langle x'|\psi \rangle = \psi(x)$, where $\langle \phi|\psi \rangle $ is the dot product in $L^2(\mathbb{C})$. I found out, that this is true, if you set x' to ...
2
votes
1answer
629 views

Any even elliptic function can be written in terms of the Weierstrass $\wp$ function

Given two nonzero complex numbers $\omega_1, \omega_2$, with nonreal ratio, we define the period module $$M= \omega_1 \mathbb Z+ \omega_2 \mathbb Z= \{n_1 \omega_1+ n_2 \omega_2:n_1,n_2 \in \mathbb Z ...
5
votes
1answer
150 views

Prove that an eigenvector is the maximum of a symmetric matrix

Let $f : S^{n-1} \rightarrow \mathbb{R}, x \mapsto x^TAx$ ( A is a symmetric matrix), then an eigenvector $\xi$ of A is a local maximum of this function. We are supposed to prove this in 6 steps and ...