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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0answers
184 views

isometric isomorphism between normed spaces and its dual

Let $E$ and $F$ be normed spaces. If $E \equiv F$ (isometry isomorphic), Does $E^* \equiv F^*$ (isometry isomorphic)? Where $E^*$ and $F^*$ are continuous dual spaces.
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1answer
56 views

Convergence of $\sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2}$

Does the series $$ \sum_{n=1}^\infty\left(1+\frac{2}{n}\right)^{n^3+n^2+1} \mathrm{e}^{-2n^2} $$ converge? The ratio test is inconclusive, so I think I must use the comparison test. But I couldn't ...
1
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3answers
71 views

Prove $\frac{n^2+2}{(2 \cdot n^2)-1} \to \frac{1}{2}$

Prove $\frac{n^2+2}{(2 \cdot n^2)-1} \to \frac{1}{2}$ for $n \to \infty$. I've been looking at this for hours! Also, sorry I don't have the proper notation. This is where I'm at: $$ \left| ...
2
votes
1answer
34 views

Closure and subbasis

Let $X$ be a topological space and $A \subset X$ with a subbasis $S$. Does it then hold that $x \in \overline{A}: \Leftrightarrow \forall s \in S: (x \in s \Rightarrow s \cap A \neq \emptyset).$ This ...
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2answers
60 views

Question analysis so amazing [duplicate]

How this process has been calculated in a manner added.
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1answer
31 views

Give example where an outer measure is strictly less than the set function from which it is defined.

Let $K $ be a class of subsets of $X $ where for every subset $A $ of $X $ there is a sequence $\{E _n \} $ of sets in $K $ such that $A \subset \bigcup _{n=1 }^{\infty } E _n $. Let $\lambda$ be a ...
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1answer
60 views

Reparametrization of a curve which is not regular

Let $\alpha : [a,b] \rightarrow \mathbb R^3$ be a $C^1$ mapping (curve). Then $\alpha$ has a length. If $\alpha'(t)\neq 0$ for all $t\in [a,b]$ then, denoting $$ \sigma(t)=\int_a^t |\alpha'(u)|du, $$ ...
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5answers
121 views

If $f=u+iv$ is an entire function such that $u^2\geq v^2,$ then $f$ is constant

Let $f=u+iv$ be an entire function such that $u^2(z) \geq v^2(z), \forall z \in \mathbb{C}.$ Could anyone advise me how to prove $f \equiv$ constant $?$ Hints will suffice. Thank you.
2
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1answer
33 views

Proving that an analytic function that maps on to {$z\in \mathbb{C}| |z-2|=1$} from some connected open set is constant

This is the approach I took to solve this but I got stuck. Suppose$f=u+iv\in $ {$z\in \mathbb{C}| |z-2|=1$} and that $f$ is analytic on an open connected set. Then we have that $(u-2)^2+v^2=1$. ...
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0answers
47 views

Analytic continuation Dirichlet series

I have a Dirichlet series $A(s)$ with an absolutely convergent Euler product for $\sigma >0$. The zeros of the factors converge to $0+2\pi k$. I now have to proof that there can't be an analytic ...
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1answer
38 views

On Pseudometric

How a pseudometrics induces topology? Can anyone discuss on this topic or give any good reference?
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1answer
63 views

If $f$ is differentiable everywhere, is $f'$ the weak derivative?

Let $f \in C^0([0,T])$ be such that $f'$ exists in the classical sense everywhere, but $f'$ may not be continuous. Is it true that $f'$ is the weak derivative of $f$ too, if it exists? I know this is ...
0
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1answer
59 views

Exam Question from multivariable calculus.

This question from a previous multivariable calculus exam.I don't know how to start with this question: Let $f$ be differentiable at every point of line segment joining $x_0$ and $x_0+h$.Show that ...
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1answer
15 views

What conditions have to be met for a subset $A$ of a measurable set $X$ to be also measurable?

What conditions have to be met for a subset $A$ of a measurable set $X$ to be also measurable? I understand that the union of measurable of sets is also measurable. But I am wondering if there is ...
0
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1answer
29 views

I do not understand why $E_n$ is measurable in this proof

Suppose we have a sequence of measurable functions $\{f_n\}$ on $X$. Also, suppose that (a) $0\leq f_1(x)\leq f_2(x)\leq ...\leq\infty$ for every $x\in X$ (b) $f_n(x)\rightarrow f(x)$ as ...
1
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1answer
42 views

Hölder norm bounded by $L^p-$norm?

Let $C_0^{\alpha}(\mathbb{R})$, $0<\alpha<1$ denote the space of Hölder-continuous functions on $\mathbb{R}$ with compact support. Is it true that for any $f\in C_b^{\alpha}(\mathbb{R})$ one ...
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3answers
42 views

sum of two closed sets

Sum of two closed sets in $\mathbb{R}^2$ need not be closed. If one of them is compact, then the sum is closed. But if I'm given any two closed sets in $\mathbb{R}^2$, then how do I check whether it ...
3
votes
4answers
316 views

Is it true that every bounded sequence with the following property converges?

Is it true that every bounded sequence $\{a_n\}$ of real numbers such that $|{a_n - a_{n-1}}|<1/n$ for all $\ge2$ is convergent?
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4answers
68 views

Let $E \subset \mathbb{Z}$ be non-empty, bounded below. Prove that $\inf{E} \in E$

I know that since $E$ is non-empty and bounded below that $\inf{E}$ exists, but I'm not sure how to show that it is in $E$.
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1answer
85 views

Constructing a countable dense subset of a totally bounded set

Given a metric space $(X,d)$, and (non-empty) totally bounded set $E$ in $X$, is it possible to construct $D \subseteq E$ which is countable and dense? I feel that this should definitely be possible. ...
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1answer
36 views

Is $\Omega$ not open?

Suppose $\Omega\subset\mathbb{R}^d$ is connected. Let $z:[0,1]\to\Omega$ be a continuous path. Suppose $\underset{t\in[0,1]}{\inf}\text{dist}(\partial\Omega,z(t))=0$. Is $\Omega$ necessarily not ...
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2answers
79 views

Subject GRE question - set of points of discontinuity

I was just working on a Math Subject GRE practice test, and I got the following problem wrong: Let $f$ be the function defined on the real line by $\displaystyle f(x) = \begin{cases} \displaystyle ...
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2answers
40 views

Is this correct the rational numbers?

Determine the rational numbers $a,b$ , if , $($$2a-b$$)$ - $2b\sqrt3$ $=$ $3$ + $2\sqrt3$ I'm thinking that $-2b\sqrt3$ = $2\sqrt3$ $=>$ $ b = -1 $ $2a-b$ resembles with $3$ , and i solved ...
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0answers
49 views

Bounded variation in the context of Feller's paper on Muntz' Theorem

The paper I have posted a picture of is a paper of Feller. He shows that the functions $f_k$ are Laplace transforms of $C^\infty$ functions $u_k$. In order to execute his suggested proof, I ...
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1answer
55 views

Is the number of subsequential limits of a sequence always countable

I know that a sequence can have many different subsequential limits but is the number of subsequential limits always countable? How do we know?
2
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1answer
115 views

Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$?

Is every Lipschitz continuous function is holder continuous with exponent $\in (0,1)$? This seems to be true,but I haven't found such a conclusion in any textbook.
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2answers
61 views

Prove the existence of a greatest lower bound of $X$ if $X \subset \mathbb{R}$ is a non-empty set that is bounded below

Attempt: Let $C \subset \mathbb{R}$ be the set of all lower bounds of $X$. Since $C$ is not empty and bounded above, every $x \in X$ is an upper bound of every element $c \in C$. Thus, there exists ...
1
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1answer
40 views

$\partial(S') \subset \partial S$ iff $S' \cap S^o \subset (S')^o$

Usually I can come up with some ideas but this time I don't. It would be great if you can tell me how I would make use of the first part of the question to prove the equivalent relation. Question: ...
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1answer
65 views

partial derivative of $f(x,y)$ who satisfies $f_{xx}-f_{yy}=0$

Suppose that $z=f(x,y)$ and its second-order partial derivative is continuous. It also satisfies $\displaystyle\frac{\partial^{2}f}{\partial x^{2}}-\frac{\partial^{2}f}{\partial y^{2}}=0$,$f(x,2x)=x$ ...
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votes
1answer
34 views

A simple question about the laplacian

Suppose that $f:X\subseteq R^n\to R$ depends only on the distance $(x_1,x_2,...,x_n)$ is from the origin in $R^n$ (i.e. $f(\vec x)=g(r)$ where $r=\left | \vec x \right |$) Show that for all $\vec x\ne ...
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3answers
77 views

Is $f(x)=x$ the solution of an integral equation? [closed]

Suppose that $f:[0, \infty)\longrightarrow \mathbb{R}$ is continuous and $f(x) \neq 0 $ for all $x>0$. If $$ \big(\,f(x)\big)^2=2 \int_0^x f(t)\,dt, $$ for all $x>0$, is it then true that ...
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1answer
46 views

Find the cartesian equation for $(e^t,t^2)$

This isn't one I recognise. I want to express it as $f(e^t,t^2)=c$ (a level curve) but I'm not sure how. I have arrived at a partial derivative equation (knowing that in the direction $(e^t,t^2)$ ...
2
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0answers
38 views

Non-separability of the Cadlag functions equipped with the topology of uniform convergence on compacts

Consider the space $D:=D([0,\infty),\mathbb{R}^N)$ of component-wise right continuous functions with left limits. Endow $D$ with the metric ...
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1answer
42 views

Sign of the eigenvalues of the Laplacian

I have to prove that, given the problem$$ \begin{cases} \Delta\:g+ \lambda \:g=0\quad {\rm in}\;D \\ g=0\quad {\rm on} \; D\end{cases}$$ then the eigevalues $\lambda>0$. I multiply the first ...
2
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1answer
42 views

Solution transport equation

I have to solve the following equation $$\partial_t v+2A\cdot\nabla v-iB(x)\cdot Av=0$$ where $A$ is a constant vector and $B$ a smooth vector field. I can solve the transport equation $\partial_t ...
0
votes
1answer
65 views

Whether there is a continuous bijection from $(0,1)$ to closed interval $[0,1]$. [duplicate]

Is there a continuous bijection from open interval $(0,1)$ to $[0,1]$. The answer is not. How to prove? I think it may proceed by contradiction and apply open mapping theorem. However, $(0,1)$ is ...
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0answers
23 views

“Interpolating between estimates”?

the headline reproduces the whole problem. What is meant by saying "Interpolating between the estimates (A) and (B), we finally obtain..."? For beeing mor specific I'll give the concrete estimates ...
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1answer
88 views

About Riemann integrability

I need to prove if $f$ is continuous on an interval $I$, then its Riemann integral exists. It is hard for me because it is an interval and not closed interval. Can anyone give me some answers or ...
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votes
1answer
41 views

Condition for existence of a continuous function

Let $ {{x_n}} $ be a given sequence. Show there exist a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that $f(1/n)=x_n$, if and only if $x_n$ converges to a finite limit.
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1answer
29 views

Is $-\log(x^2) \geq 2(1-x)$ for $x \in (0,1]$?

I want to show that $-\log(x^2) \geq 2(1-x)$ for $x \in (0,1]$, where $\log$ is the logarithm to base 2. How can I do that? I tried to make an estimate by first bringing the minus to the other side ...
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1answer
23 views

I really need help to fill the gaps on one statement in completion of a measure theorem´s proof (edited question)

Let $(X,S,\mu)$ a measure space and $N =\{N' \in S : \mu(N')= 0 \}$ the $\sigma$-ring of $S$-measurable sets of measure zero. We define : $S^*= \{(E\cup M_{1})-M_{2} : E\in S \text{ and } M_{1} ...
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0answers
20 views

I really need help to fill the gaps on one statement in completion of a measure theorem´s proof

Let $(X,S,\mu)$ a measure space and $N =\{N' \in S : \mu(N')= 0 \}$ the $\sigma$-ring of $S$-measurable sets of measure zero. We define : $S^*= \{(E\cup M_{1})-M_{2} : E\in S \text{ and } M_{1} ...
0
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1answer
19 views

How to alter the interval of a composite function

Let $f, g : R → R$ $$f(x) = \begin{cases} x + 3 &\text{if } x ≥ 0,\\ x^2 &\text{if } x < 0 \end{cases}$$ $$g(x) = \begin{cases} 2x + 1 &\text{if } x ≥ 3,\\ x ...
1
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1answer
96 views

Prove the following function is differentiable

I have to prove if this function is differentiable. $$f(x,y)= \begin{cases} (x^2+y^2) \sin\frac 1{(x^2+y^2)} \iff (x,y) \neq (0,0) \\0 \iff (x,y)=(0,0) \end{cases}$$ I tried proving that all of its ...
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2answers
100 views

Chapter 3, question 20 part b of Spivak's Calculus 3rd edition

Suppose that $f(y)-f(x)\le(y-x)^2$ for all $x$ and $y$. (Why does this imply that $\lvert f(y)-f(x)\rvert \le (y-x)^2$ ?) .Prove that $f$ is a constant function. Hint: Divide the interval from $x$ to ...
11
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1answer
222 views

Is there always an equivalent metric which is not complete?

I have seen that completeness is not a topological property like compactness or connectedness. I have seen some examples also showing that there are two equivalent metrics one of which is complete and ...
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1answer
32 views

$2^{\mathrm{nd}}$ order nonlinear ODE: $4y''\sqrt{y}=1, y(0)=1, y'(0)=1$

I am solving this second order nonlinear equation, that is in the title. My solution is: $$ \frac{4}{3}(y^{1/2}+c)^{3/2}-4c(y^{1/2}+c)^{1/2}+a=x $$ where $c$ and $a$ are constants that spawned ...
3
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2answers
90 views

Lebesgue measure of any line in $\mathbb{R^2}$.

What is the Lebesgue measure of a line in $\mathbb R^2$? I am guessing that this zero. But i couldn't prove it rigorously. Please help... From this can i conclude that any proper subspace of $\mathbb ...
0
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0answers
30 views

Convergence of probability density in the tail.

Let $f(x)$ be a probability density with respect to the Lebesgue measure. The distribution has first moment, e.g. $\int_{-\infty}^\infty |x| f(x) dx < \infty$. Further assume that there exists $K$ ...
0
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2answers
44 views

How to show $f$ is a constant zero function…

Power series $f(x)=\sum\limits_{n=0}^\infty$ $a_n x^n$ with the radius of convergence $R>0$ And the sequence $(b_k)$ satisfies $R>b_1>b_2>..., $ $\lim_{k\to\infty} b_k=0$ Need to show that ...