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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2
votes
2answers
31 views

Does $ \sum_{n=0}^{\infty} a_n |\ln a_n| < +\infty$?

Assume $\sum_{n=0}^{\infty} a_n < +\infty$ and $a_n > 0$. dose $$ \sum_{n=0}^{\infty} a_n |\ln a_n| < +\infty$$ hold?
5
votes
2answers
89 views

Prove that $\displaystyle\prod_{q\in \mathbb{Q}^{\times}}|q|=1$

Prove that $\displaystyle\prod_{q\in \mathbb{Q}^{\times}}|q|=1$. I don't have a lot of experience working with infinite products, but I read a couple of theorems that say that absolute convergence of ...
0
votes
0answers
35 views

Boundedness in dimension 1 and 2.

Consider $\psi$ in $\mathbb{R}^m$ such that $$\lim_{|x|\rightarrow\infty}\frac{|\psi(x)|}{|x|^{1-m/2}}\leq C,$$ where $C$ is a positive constant. Why in dimensions 1 and 2 is sufficies to assume that ...
0
votes
1answer
70 views

Question about boundness of derivatives.

My doubt is in the paper: Further qualitative properties for elliptic equations in unbounded domains, by Berestycki, Caffarelli and Niremberg (page: 93) My question is simples. For any direction ...
2
votes
1answer
221 views

Rational Function Theorem related to Integration?

I know how to use this algorithm when I am integrating rational functions, but my textbook has omitted the actual proof for why it works. If someone could please help me with this question:
1
vote
2answers
125 views

$\sin x$ does not satisfy this quadratic equation

Prove that $\sin x$ is not a rational function using the fact that it is not of the form $p(x)/q(x)$ where $p$ and $q$ are polynomials. Then, by using the above proof, prove that $\sin x$ does not ...
1
vote
1answer
79 views

Find a majorizing function

Please, could somebody help me find a function $f(x)$ such that $| \frac{1}{n+n^2 \sin(xn^{-2})}| \le f(x)$ for each $n \in (0, \infty)$. $f(x)$ has to be $\ge 0$ for every $x \in (0, \infty)$ and ...
1
vote
1answer
55 views

Analysis convergence

This is a past exam question which im not sure how to solve.. ${For \ n=1,2,3,... consider \ f_n: [0,2] \rightarrow \mathbb{R} \ given \ by}$ $f_n(x)=$ $\left\{\begin{array}{l l}nx & \quad ...
0
votes
2answers
63 views

Trigonometric proof query

I am having trouble proving the following identity (where $m,n \in \mathbb{R}$ are arbitrary): $$\sin(mx)\sin(nx) = \frac{1}{2}[\cos(m -n )x - \cos(m + n)x] \quad (1)$$ By expanding the RHS, I can ...
0
votes
1answer
203 views

Analysis.. Norm on C([a,b])

Let $w:[a,b]\rightarrow \mathbb{R}$ with $ w(x)\geq c>0 $ for some $c \in \mathbb{R}$ and all $x \in [a,b]$. Prove that $$\lVert f\rVert_w \ = \ \displaystyle\int^b_a \lvert f(t)\rvert w(t)\ ...
0
votes
1answer
82 views

Analysis.. Convergence of sequence

I really struggle with understanding convergence and have the following questions.. Determine whether the following sequences converge and if so, give the limit: $x_n = ...
0
votes
2answers
176 views

Product of limsup

Let $f(x)$ be positive and increasing and $g(x)$ satisfy $\limsup_x g(x)=1$. I want to show $\limsup_x f(x) g(x)=\infty$ Is that true and how do i show it? I'm thinking that since $f(x)$ is ...
2
votes
2answers
180 views

If $f_n(x)=x^n$ converges to $f$, why is $f$ not continuous?

I was reading my Analysis course notes and had some trouble. I hope you can help me. Let $C(X)=\{ f | f:X \longrightarrow \mathbb{R} \text{ is a continuous function}\}$. It was already stated and ...
2
votes
1answer
42 views

Anharmonic series, find a permutation of its indices so that its sum is 0

Could you help me solve this problem? Find such a permutation of indices of anharmonic series ($\sum _{n=1}^{\infty} (-1)^{n+1} \frac{1}{n}$ ) so that after this permutation its sum equals $0$.
4
votes
1answer
122 views

Polynomial expression of $\frac{\sin x}{x} $

Could you explain to me why $$\frac{\sin x}{x} =\left(1-\frac{x^2}{\pi ^2}\right)\left(1-\frac{x^2}{(2 \pi) ^2}\right)\left(1-\frac{x^2}{(3 \pi )^2}\right)\cdots$$ I've read in this article ...
2
votes
1answer
71 views

Is multiplying by a measurable function $V$ always self-adjoint?

There are a handful of results establishing conditions on the measurable real-valued function $V(x)$ under which the operator: $$-\Delta + V(x)$$ Is (essentially) self-adjoint on ...
3
votes
1answer
240 views

Integration of hyperreal functions / Intermediate Value Theorem

Here's a statement on hyperreal function I've been trying to prove (I came up with it but I think it is true): Suppose $f(x)$ is a continuous real-valued function and $h(x)$ is a continuous ...
1
vote
2answers
188 views

Show vector (p-q) is orthogonal to the curve at q

Let $f:\mathbb R \to \mathbb R^n$ be a differentiable mapping with $f'(t) \neq 0$ for all $t$ in $\mathbb R$. Let $p$ be a fixed point not on the image curve of $f$. If $q = f(t_0)$ is the point of ...
2
votes
1answer
151 views

Equivalent statements of continuity of linear operators

I am asked to prove that the following are true: Given a linear operator $T: X \to Y$ where $X,Y$ normed linear spaces: (1) $T$ continuous at at point $\iff$ $T$ continuous everywhere (2) $T$ ...
1
vote
1answer
62 views

Show that f '(a) exists for all a if f is linear

If $f: \mathbb R \to \mathbb R^m$ is linear, prove that $f'(a)$ exists for all $a$ in $\mathbb R$, with $dfa = f$
1
vote
1answer
89 views

Sequential continuity in normed linear spaces

I am trying to prove the following "contiuity-type" result. Let $X,Y$ normed linear spaces. Let $\{T_n\} \to T \in \mathcal{L}(X,Y)$ and $\{u_n\} \to u \in X$. Show that $\{T_n(u_n)\} \to \{T(u)\} ...
3
votes
0answers
135 views

Singular derivative matrix implies not one to one?

I am trying to show that if $f:\mathbb{R}^n\to \mathbb{R}^n $ is continuously differentiable and that for all $x\in \mathbb{R}^n$ $Df(x)$ is singular implies that $f$ is not 1-1. The singularity of ...
1
vote
1answer
480 views

Integration of even (and odd) function

Suppose that $a>0$ and that $f$ is integrable on $[-a,a]$. Show that if $f$ is even then $$ \int_{-a}^0 fdx = \int_0^a fdx $$ using the Riemann sum definition of Riemann integrability. This is ...
0
votes
1answer
28 views

Directions of decrease for a convex functions

Suppose $f(x,y)$ is a convex function and $$ f(x+\Delta x, y) < f(x,y), ~~~ f (x, y + \Delta y) < f(x,y)$$ Does this imply $$ f(x+\Delta x, y + \Delta y) < f(x,y)$$? I am guessing the ...
1
vote
1answer
210 views

Is there a simple smooth non-piecewise function that could replace this piecewise one

I need a function which satisfies these conditions: $f(0) = 0,$ $f(x)$ is monotonically increasing for all $x > 0,$ for some specified $x_2 > x_1 > 0$ and $y_2 > y_1 > 0,$ $f(x_1) = ...
4
votes
1answer
357 views

Image of unit ball dense under continuous map between banach spaces

I am assuming that the following problem will require the open mapping theorem, or maybe the closed graph theorem. Any help that can be given will be deeply appreciated. The statement is the ...
4
votes
2answers
122 views

Normed linear space and linear functional

Let $X$ be the normed linear spaceof sequences of reals that have only finitely many non-zero terms. Given $x = \{x_n\} \in X$, define $$f(x) = \displaystyle \sum_{n=1}^{\infty} x_n$$ I think that it ...
3
votes
1answer
73 views

Inequality concerning nonnegative numbers (related to Hanner's inequalities)

I've been having a look at how Lieb and Loss (in their textbook on Analysis) prove Hanner's inequalities and have been trying to get a handle of the geometric intuition involved. In doing so I've been ...
1
vote
0answers
60 views

Normed space Analysis

Show that the normed space $(\ell^2, \|\cdot\|_2)$ is complete. I am not sure where to even start with this question. I'm quite sure it involves Cauchy and also Bolzano-Weierstrass but I'm not sure ...
4
votes
2answers
667 views

Metric Spaces Analysis

Let $(X,d)$ be a metric space and for $x,y \in X$ define $d_b(x,y) =$ $ \dfrac{d(x,y)}{1 + d(x,y)}$ a) show that $d_b$ is a metric on $X$ Hint: consider the derivative of $f(t)$ = $\dfrac{t}{1+t}$ ...
2
votes
2answers
78 views

Norms Abstract Analysis

I have a question relating to norms and have been giving functions and need to state whether they are norms or not... which of the following are norms on $\mathbb{R}^2$? Give reasons for your ...
2
votes
0answers
157 views

What is the total variation measure of the integration of a kernel of signed measures?

Assume given a probability space $(\Omega,\mathcal{F},P)$ and a measurable space $(E,\mathcal{E})$. Let $(\nu_\omega)_{\omega\in\Omega}$ be a family of signed measures on $(E,\mathcal{E})$. Assume ...
6
votes
4answers
1k views

Using Taylor's Theorem to show that $\ln(1 + x^2) \leq x^2$

Can we show that if $\operatorname{abs}(x) \lt 1$, then $$\ln(1+x^2) \leq x^2\;,$$ using Taylor's Theorem? I am thinking of expanding it about $x=0$ but I got something like $$f(x) = -x^2 + ...
0
votes
1answer
203 views

Question of Hausdorff measure

I'm self-learning graduate Real Analysis and currently on Hausdorff Measure section. Can someone help me with this question? Thank you so much. Consider $X=\mathbb{R}$. Prove that for any subset ...
0
votes
0answers
56 views

Real Analysis Hausdorff measure

Can someone help me with this Hausdorff measure question. I'm self learning graduate Real Analysis and I don't know how to type notations so I just take pic of the question.
2
votes
3answers
273 views

Limit Question - Explanation

The limit of $f(x) = x$, as $x$ tends to zero is zero. What's the limit of the function $\dfrac{x^2}{x}$ as $x$ tends to zero? and What's the limit of the function f(x) = (modulus of x)/x ? I am ...
1
vote
0answers
53 views

Trouble proving/disproving the existence of a particular derivative.

I am having trouble with the following question: Quesiton: Define $f$ such that $$ f(x) = \begin{cases} \cos\left(\frac{1}{x}\right) &\mbox{ if }x \ne 0 \\ 0 &\mbox{ if } x = 0 \end{cases} ...
1
vote
2answers
142 views

Closure in Metric Space

I need help understanding Theorem 2.27(c) in Rudin. If $X$ is a metric space and $E \subset X$, and if $E'$ denotes the set of all limit points of $E$ in $X$, then the closure of $E$ is the set $\bar ...
0
votes
1answer
344 views

Discrete approximation - exponential function and integrals

Let $f$ be a complex-valued continuous function on $\mathbb{R}_+$ with compact support and let $g, h$ be two complex-valued continuous functions on $\mathbb{R}_+$ such that $g$ is bounded and ...
7
votes
3answers
289 views

Even integer approximations to multiples of pi

I admit that I'm probably out of my depth with this question, but I can't help but feel curious. I wanted to show that, in the sequence $\{\sin(n)\}$, there is never a largest term (the sequence ...
2
votes
1answer
41 views

A simpler proof that this series converges

I am given that $a_n \ge 0$ and $\sum a_n$ converges. I need to show that $\sum \frac{\sqrt{a_n}}{n}$ converges also. After a long time, I came up with a solution by re-ordering the series into: ...
2
votes
1answer
40 views

Is this equation valid $\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$,?

While reading a script I found this equation: $\gamma b^{e \log_b{n+e}} = \gamma b^e + n^e$ and i cannot figure out how the author did this. I'd appreciate a step-by step equation for this ...
0
votes
1answer
262 views

A change of variables in the euler equation

If someone could help me with the proposed change of variables, it would be greatly appreciated. Consider Euler's equation: $$z^2w'' + \alpha zw' + \beta w = 0$$ where $w$ is a function of $z$ and ...
1
vote
1answer
210 views

Legendre's equation polynomial solution

This is a problem on analytic solutions of ordinarry differential equations. Any help will be greatly appreciated. Please, try to be as specific as possible as I don't handle this material very well ...
3
votes
2answers
348 views

Analysis proof for repeating digits of rational numbers

"Every rational number is either a terminating or repeating decimal". I knew there's a proof for this using number theory's theorems, but I wish to find a purely analysis proof, that is: the series ...
3
votes
2answers
51 views

Application of FTC and change of variable

Let $f:[0,1]\to \mathbb{R}$ be continuous such that $$\int_{0}^{1} f(xt)dt=0$$ for all $x \in [0,1]$. Show that $f(x)=0$ for all $x \in [0,1]$. Using the FTC and substitution: ...
3
votes
3answers
201 views

Show this sequence is equicontinuous

I'm stuck on an analysis problem to which I've reduced to the following, so some assumptions may be superfluous. Let $\{ f_n(x) \} \subset C(X,\mathbb{R}^{\geq0})$ (i.e. $f_n$ is continuous and ...
1
vote
2answers
125 views

Convergence of a particular double series

For the double series $$ \sum_{m,n=1}^{\infty} \frac{1}{(m+n)^p} , $$ I was wondering when it converges. I want to use double integrals to estimate it, but I don't know how to write the process ...
1
vote
1answer
468 views

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$.

Show that a map $f:X\to Y$ is onto iff $f(f^{-1}(C))=C$ for all subsets $C\subseteq Y$. My workings so far: Because this is an if and only if proof we need to show it both ways. First let's assume ...
3
votes
2answers
146 views

Elementary application of Brouwer's fixed point Theorem

A professor of mine has suggested to me to look at this theorem and to find a problem related to it to explain in a future class. I found an understandable proof in "Linear operators" by ...