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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4
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1answer
199 views

Evaluating a limit involving binomial coefficients.

If $N_c=\lfloor \frac{1}{2}n\log n+cn\rfloor$ for some integer $n$ and real constant $c$, then how would one go about showing the following identity where $k$ is a fixed integer: $$\lim_{n\rightarrow ...
5
votes
2answers
329 views

Isomorphism of Banach spaces implies isomorphism of duals?

I can't make up my mind whether this question is trivial, or simply wrong, so i decided to ask, just in case someone sees a fallacy in my reasoning: Question: Suppose $V,W$ are two banach spaces, and ...
2
votes
1answer
616 views

Mutually Singular measures

c.f. Rudin's Real and Complex Analysis (Third Edition 1987) Chapter 6 Q9 Suppose that $\{g_n\}$ is a sequence of positive continuous functions on $I=[0,1]$, $\mu$ is a positive Borel measure on $I$, ...
3
votes
1answer
635 views

Prove: If the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then f must be a constant function

Prove: If the function $f$ is continuous on $[a,b]$, differentiable on $(a,b)$ and $f'(x) = 0$ on $(a,b)$, then $f$ must be a constant function on $[a,b]$. I need to select some $x_1$ and $x_2$ in ...
0
votes
1answer
167 views

Prove: If a function $f: (a,b) \to\mathbb{R}$ is uniformly continuous, then $f(a+)$ and $f(b-)$ are both finite.

Prove: If a function $f: (a,b) \to\mathbb{R}$ is uniformly continuous, then $f(a+)$ and $f(b-)$ are both finite. I think that the best way for this to be proved is by contradiction...
4
votes
3answers
646 views

Showing that $\sum \frac{\log n}{n^x}$ converges for $x>1$

I'm trying to show that $\sum \frac{\log n}{n^x}$ converges for $x>1$ by the ratio test. Here's what I've got so far $$\frac{a_{n+1}}{a_n} = \frac{\log (n+1) n^x}{(n+1)^x \log n}$$ ...
0
votes
2answers
340 views

Mapping a variable having very vast range to the interval (0,1)

I am trying to get a continuous mapping from (0,∞) to (0,1). What would be a good mapping? Context: I need to rank tuples based on values of two of their attributes ...
2
votes
2answers
72 views

Show $x^2$ in the interval $(0,1/3]$ has no fixed points.

Show $x^2$ in the interval $(0,1/3]$ has no fixed points. I understand that the range of that domain is always lower than $y=x$, but what is a proper way of showing this? $$\left(0,\frac13\right] ...
7
votes
1answer
913 views

Evaluating integral using Riemann sums

It is given that: $$\sin\frac{\pi }{n} \sin\frac{2\pi }{n}\cdots\sin\frac{(n-1)\pi }{n}=\frac{n}{2^{n-1}}$$ It is asked to use the above identity to evaluate the following improper integral: ...
2
votes
2answers
648 views

continuous and strictly increasing implies differentiable

I am not sure if this is true, but intuitively it seems that if a function is strictly increasing and it is also continuous...it is differentiable. It may be because there are no bumps like in the ...
4
votes
1answer
726 views

Riemann zeta function and uniform convergence

A question in a past paper says prove that this series converges pointwise but not uniformly $$\xi(x):= \sum_{n=1}^\infty \frac{1}{n^x} .$$ But I thought that it did converge uniformly to some ...
5
votes
0answers
74 views

Divergence Series [duplicate]

Possible Duplicate: Slowing down divergence 2 If $\{a_n\}$ and $\{b_n\}$ are two increasing sequences of positive numbers such that both $\displaystyle\sum_{n=1}^{\infty} \frac{1}{a_n}$ and ...
0
votes
1answer
135 views

When is this equation true?

For what kind of functions/or for which functions $h(a,b)$ is the below equation true : assumption is $\lim_{ x\to\infty }{f(x)}=l_1$, $\lim_{ x\to\infty }{g(x)}=l_2.$ When is it true that $\lim_{ ...
2
votes
2answers
544 views

Pointwise Convergence Implies Uniform?

Let $f_{n} \to 0$ pointwise on the interval $[-A,A]$ where $f_{n}$ are continuous and uniformly bounded. Can we show the convergence is uniform? Thanks, any help appreciated.
5
votes
2answers
124 views

Is it equation true - limit

Is it true that : $\lim_{ x\to\infty } \left( 1+\frac{f \left( x \right) }{x} \right) ^x = \exp \left( \lim_{ x\to\infty } f \left( x \right) \right)$ ? Assumption is that limit of $\lim_{ ...
3
votes
1answer
320 views

Discontinuity in function; order of integration matters

I'm struggling a bit with Chapter 10 of Rudin's "Principles of Mathematical Analysis," and I was hoping to get some help here. I'll post the problem and my current progress. Exercise 2: For $i = 1, ...
3
votes
2answers
146 views

Show the following series converges.

Let the real sequence ${x_n}$ be given by, $$\sum_{j=1}^{2n} \frac {1}{j} - \sum_{j=1}^{n} \frac {1}{j}. $$ Show that $0<x_{n}<x_{n+1}$ and that $x_{n}<1$ for all $n$. Deduce that $x_{n}$ ...
2
votes
1answer
249 views

Completion of a metric space

Let $R$ be a topological ring (i.e addition and product are continuous) which we assume it is metrizable with metric d and consider the completion $\hat{R}$ of the ring $R$ defined as the set of all ...
5
votes
1answer
426 views

Norm for continuous linear functionals, newbie questions

Let $E$ be a normed vector space and let $f\colon E \to \mathbb{R}$ be a continuous linear functional. Define the dual norm of $f$ as $$ \|f\| = \sup_{\|x\|\leq 1} |f(x)|. $$ First question. I ...
5
votes
1answer
4k views

Indefinite Integral for $\cos x/(1+x^2)$

I have been working on the indefinite integral of $\cos x/(1+x^2)$. $$ \int\frac{\cos x}{1+x^2}\;dx\text{ or } \int\frac{\sin x}{1+x^2}\;dx $$ are they unsolvable(impossible to solve) or is there a ...
0
votes
1answer
335 views

Support line for strictly convex function

Assume that $I$ is an interval in $\mathbb{R}$ and $f: I\rightarrow \mathbb{R}$. $f$ is called strictly convex if $$f(tx+(1-t)y) < t f(x)+(1-t)f(y)$$ for $x\neq y$, $x,y \in I$, $t\in (0,1)$. How ...
1
vote
2answers
1k views

Use the Intermediate Value Theorem to prove $f:[0,1]\to [0,1]$ continuous and $C\in[0,1]$, there is some $c \in [0,1]$ such that $f(c) = C$.

Use the Intermediate Value Theorem to prove $f:[0,1]\to [0,1]$ continuous and $C\in[0,1]$, there is some $c \in [0,1]$ such that $f(c) = C$. Using a similar technique to the proof of the ...
5
votes
1answer
256 views

Show sequence equicontinuous

I don't know how to prove this question: Let $X$ be a compact space, and let $(T_{n})$ be a sequence of positive linear operators on $C(X)$. Also, let $f \in C(X)$ be a strictly positive function. ...
6
votes
1answer
387 views

Is there a non-compact metric space, every open cover of which has a Lebesgue number?

Lebesgue lemma states that for every open cover $\{U_\alpha\}_{\alpha\in A}$ of a compact metric space $(X,\rho)$ there exists a number $d>0$ such that $$ \forall x\in X \quad \exists ...
5
votes
1answer
140 views

What is the general solution for integrals of the form $\int_{0}^{\infty}\frac{x^{2 m}\;\ln^{n}(x) }{e^{\frac{2 p +1}{2}x}} dx$?

I have this integral $$\int_0^\infty \frac{x^{2 m}\;\ln^n (x) }{e^{\frac{2 p +1}{2}x}} dx\;\; m,n,p \in \mathbb{N}$$ and I'like to know the general solution. From WolframAlpha I got some ...
1
vote
1answer
239 views

Find a sequence $\{a_n\}$ of real numbers such that $\sum a_n$ converges but $\prod (1+ a_n)$ diverges.

Find a sequence $\{a_n\}$ of real numbers such that $\sum a_n$ converges but $\prod (1+ a_n)$ diverges. The converse is trivial, just make all the $a_n=-1$.
1
vote
3answers
143 views

Proving a simple inequality.

I am trying to prove that $\frac{n}{4n^2+1} > \frac{n+1}{4(n+1)^2+1}, \forall n\in\mathbb{N}$. What I did so far was $n < n+1\\ \Rightarrow \frac{n^2}{n} < \frac{(n+1)^2}{n+1}\\ ...
1
vote
1answer
605 views

proving a function is differentiable at a point

Hi guys Im really having problems with this question: Suppose $g$ is continuous in $(-1,1)$ and differentiable in $(-1,0)\cup(0,1)$. Prove that if $\lim\limits_{x\to ...
3
votes
3answers
69 views

Why is $\int_{n}^{n+1}\int_{n}^x |f'(x)|\,dx \leq \int_{n}^{n+1} |f'(x)|\,dx$ for $x \in [n,n+1]$?

Can anyone show me how to prove the following: For $x\in \left [ n,n+1 \right ]$: $$\int_{n}^{n+1}\int_{n}^{x}\left | f'(t) \right |dtdx\leqslant \int_{n}^{n+1}\left | f'(t)\right |dt?$$
3
votes
1answer
179 views

Why is $\sum_{n=1}^{\infty }(-1)^{n+1}\frac{1}{n}.e^{-nx}$ uniformly convergent?

Why is the following series uniformly convergent:$$\sum_{n=1}^{\infty }(-1)^{n+1}\frac{1}{n}.e^{-nx}$$? where $ x\geq 0$ I tried the Weierstrass-M test, but it doesn't work here because:$\left | ...
5
votes
1answer
156 views

convergence of integral vs convergence of infinite series

Problem: Let $f\in C^{1}([0,\infty ))$ such that: $\int_{1}^{\infty }\left | f^{'}(x) \right |dx$ converges. The question is to prove the following: $\left ( \sum_{n=1}^{\infty }f(n) \right )$ ...
3
votes
3answers
958 views

Ratio test and the Root test

Both the ratio test and the root test define a number (via a limit). If both limits exist (and shows that the series is convergent), what (if any) is the relation between the 2 numbers ? are they ...
4
votes
3answers
931 views

How to prove that a continuous function is identically zero over $\mathbb{R}$?

The following problem is given at the level of a senior undergrad analysis course: We are given a continuous function $g:\mathbb{R}\rightarrow \mathbb{R}$. Assume that $\mathbb{R}$ contains a ...
18
votes
4answers
652 views

A “clean” approach to integrals.

Many fields in mathematics start from the "dirty" approach. In calculus we do all sort of $\epsilon$-$\delta$ stuffs, until topology gives an elegant formulation using open sets. A first course in ...
1
vote
2answers
102 views

Equivalent norms on $\mathbb{R}^2$

For $(\mathbb{R}^2,\|\cdot\|_2)$ and $(\mathbb{R}^2,\|\cdot\|_\infty)$ and any $x \in B((0,0),1,\|\cdot\|_2)$ how would you find a $\delta_x$ such that $B(x,\delta_x,\|\cdot\|_\infty) \subset ...
2
votes
2answers
124 views

Proving that $\|x\|_2 \geq \|x\|_1$?

How would you prove that $\|x\|_2 \geq \|x\|_1$, or in other words that $$\sqrt{\int_0^1|f(x)|^2 dx} \geq \int_0^1|f(x)|dx \quad?$$
0
votes
1answer
60 views

Extending the interval an estimate holds on by continuity

I have a question about extending the interval of an estimate using continuity. So suppose that I have positive constants $c_1, c_2, D$, some real number $r >1$ and continuous function $f(x) : ...
8
votes
5answers
2k views

limit of integral $n\int_{0}^{1} x^n f(x) \text{d}x$ as $n\rightarrow \infty$

I am trying to solve the following problem at the level of a senior undergrad analysis level. So, the problem is as follows: We are given a function $f$ which is continuous on the interval $\left [ ...
2
votes
1answer
610 views

Every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$?

We have just come across the lemma that every norm on $\mathbb{R}^n$ is equivalent to $\|\cdot \|_\infty$. My question is that do all norms on $\mathbb{R}^n$ take the form $\|\cdot \|_1, \|\cdot \|_2, ...
1
vote
0answers
160 views

Finding a force function from bodies in equilibrium

(This is an edited version of the original question, since I'm starting a bounty) I'm trying to find a function $y$ from given data. Reverse optimization, so to speak. Say we have two ...
3
votes
2answers
147 views

Proof for an Inequality

Let $e^{e^x}=\sum\limits_{n\geq0}a_nx^n$, prove that $$a_n\geq e(\gamma\log n)^{-n}$$ for $n\geq2$, where $\gamma$ is some constant great than $e$.
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vote
3answers
184 views

Showing a series converges non-uniformly

For the series $$f(x) = \sum_{n=0}^\infty \frac{1}{n(1+nx^2)}$$ my lecture notes use the Weierstrass M-test to show that this converges uniformly on any interval of the form $(-\infty,-a)$ or ...
0
votes
2answers
294 views

Pointwise and uniform convergence

Can a series converge neither pointwise nor uniformly, or are they the only two 'options' for convergence? Clearly uniformly $\implies$ pointwise, but can a series be neither?
1
vote
1answer
324 views

Weierstrass M-test for uniform convergence

With the Weierstrass M-test for uniform convergence where for a functions $f_r:A \to \mathbb{R}$ you find an $M_r \in \mathbb{R}$ where $\forall x \in A, |f_r(x)| \leq M_r $. Can the $M_r$ be also a ...
2
votes
1answer
120 views

Continuous at some and at all points

Can I have someone to show me an insight on how to prove these? I had referred to a number of books but most authors merely state them as definitions or theorems without proof. Let $T:X \to Y$ be a ...
4
votes
1answer
144 views

A simple application of Hölders inequality (I think)

I'm reading a paper where the following inequality appears. $$ \| \widehat{f} \|^2_{L^2(d\mu)} \leq \| f \ast \widehat{\mu} \|_p \| f \|_{p^\prime} $$ where $f$ is a real-valued measurable function on ...
1
vote
1answer
100 views

compact nonempty sets contain Sups.

If $E$ iis a compact, nonempty subset of REAL numbers (hence closed), we know that every convergent sequence in $E$ converges in $E$, are there sequences $a_{n}$ and $b_{n}$ in $E$ that converge to ...
1
vote
0answers
66 views

On convergence $f*\phi_c$ to $f$ in $L^\infty$

Let $\phi \in L^1(\mathbb{R})$ be such that $\int_{-\infty}^\infty \phi(x)dx=1$ and let, for $c>0$, $\phi_c(x)=\frac{1}{\varepsilon} \phi(\frac{x}{c})$. In the book Introduction to Fourier ...
2
votes
0answers
124 views

norms on a vector space - is there a quicker way to approach this problem?

I'm looking at the following problem, which I have done but wonder whether there isn't a faster way of doing it. (It's a past exam question which is supposed to take 7.5 minutes, but I only managed to ...
0
votes
4answers
335 views

Help finishing a proof about compact/connected sets.

Original Question: Suppose that $X$ and $Y$ are metric spaces and that $f:X \rightarrow Y$. If $X$ is compact and connected, and if to every $x\in X$ there corresponds an open ball $B_{x}$ such that ...