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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51 views

What is the answer of this problem?

Suppose that $f(x)$ is bounded on interval $[0,1]$, and for $0 < x < 1/a$, we have $f(ax)=bf(x)$. (Note that $a, b>1$). Please calculate $$\lim_{x\to 0^+} f(x) .$$
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1answer
209 views

definition of total variation of a complex measure does not depend on any algebra generating the sigma-algebra of that measure

While studying a course on "Vector Measures", I come to this problem: Let $\mu$ be a complex measure on a $\sigma$-algebra $\Sigma$, generated by an algebra $\mathcal{A}$. Its total variation $|\mu|$ ...
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1answer
70 views

Many points on hyperplane with probability zero

Let $m$ be a finite measure on $X \subseteq \mathbb{R}^n$, so that $m(\mathbb{R}^n) < \infty$. Define the hyperplanes on $\mathbb{R}^n$, parametrized by $A \in \mathbb{R}^{n \times n}$ and $b \in ...
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43 views

If $N$ is a complete linear subspace of $H$ and $\inf_{f \in N} |f-g| = d$, prove that $\exists f'$ such that $f' \in N$ and $|f'-g|=d$

The problem as stated is Let $H$ be a Hilbert space. If $N$ is a complete linear subspace of $H$ and $\inf_{f \in N} |f-g| = d$, prove that $\exists f'$ such that $f' \in N$ and $|f'-g|=d$. I ...
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130 views

How prove this analysis function $a\le\frac{1}{2}$

let $$f(x)=\begin{cases} x\sin{\dfrac{1}{x}}&x\neq 0\\ 0&x=0 \end{cases}$$ show that:there exsit $M>0,(x^2+y^2\neq 0)$ , $$F(x,y)=\dfrac{f(x)-f(y)}{|x-y|^{a}}|\le M ...
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2answers
185 views

Proving it doesn't exist a homeomorphism between $\mathbb R$ and $\mathbb R^n$, $n>1$.

I have to prove that for $n \ge 2$, there doesn't exist a homeomorphism between $\mathbb R$ and $\mathbb R^n$. Could anyone give me a hint on how could I prove this?
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99 views

What is an example of an open set in $\mathbb{R}^2$ which is a Cartesian product of two non-open sets in $\mathbb{R}$?

What is an example of an open set in $\mathbb{R}^2$ which is a Cartesian product of two non-open sets in $\mathbb{R}$? Is this possible? I can only come up two open sets in $\mathbb{R}$? i.e (0,1) x ...
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73 views

Prove that lim is 0.

Prove that $$\lim_{x\to0}\sqrt{|x|}\sin\left(\frac1x+x^{10}\right)=0.$$ How do I show in a rigorous way that this limit as $x\to 0$ equals $0$ ? Any tips or suggestions would be great!
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52 views

Prove that this limsup inequality is strict.

so they are non-negative and bounded. how do i show this?
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1answer
41 views

Application of Stirling's theorem for the given series

I want to prove whether $x=-4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$ I applied alternating series test. But, while using this, I need to apply Stirling's ...
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158 views

Prove that lim sup $x_n$ for infinitely many n.

Let $\{x_n\}$ be a real sequence and $r$ a real number. Prove that $\lim \sup_{n\to\infty}x_n<r$ implies $x_n<r$ for $n$ large enough. Prove that $\lim \sup_{n\to\infty}x_n>r$ implies ...
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109 views

Show $e^{D}(f(x)) = f(x+1)$ where $D$ is the derivative operator

I would appreciate help showing $e^{D}(f(x)) = f(x+1)$ Where $D$ is the linear operator $D: \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ where (in the context where this statement arose) $x \in ...
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1answer
28 views

The convergence interval of the series

I want to prove whether $x=4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$ I used Raabe's test. But I got limit is 1. So the test is not valid. Please help me ...
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3answers
286 views

Family of connected sets, proving union is connected

I am having some trouble trying to prove the following statement:$$$$ Let $(X,d)$ be a metric space and $\mathcal A$ a family of connected sets in $X$ such that for every pair of subsets $A,B \in ...
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0answers
221 views

The solution of the contour integral for $\epsilon =+1$

I understand the solution for $\epsilon =-1$. And I am trying the solve this question for $\epsilon =+1$. This is important for me. I want really to learn perfectly because I am continuously seein' ...
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4answers
445 views

Prove that if all the vertices of a graph have degree 3, then the graph must have a cycle

Hello can you help me to prove this. The hint for the problem is: Think of what it means for a graph to have no cycles. So I believe this will be a contrapositive proof, but still could not do it.
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2answers
53 views

Show that $f: GL_{n}(\Bbb R) \to GL_n \Bbb (R): A \mapsto A^{-1}$ is infinitely differentiable

Consider the continuous image $f: GL_{n}(\Bbb R) \to GL_n \Bbb (R): A \mapsto A^{-1}$ I'm trying to proof with induction that $f$ is infinitely differentiable. I now understand how I can proof that ...
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1answer
166 views

definition of “weak convergence in $L^1$”

I have encountered two definitions of weak convergence in $L^1$: 1) $X_n\rightarrow X$ weakly in $L_1$ iff $\mathrm{E}(X_n\mathrm{1}_A)\rightarrow \mathrm{E}(X\mathrm{1}_A)$ for every measurable set ...
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2answers
90 views

Iteration of an operator

Let $f_0(x)$ be integrable on $[0,1]$, and $f_0(x)>0$. We define $f_n$ iteratively by $$f_n(x)=\sqrt{\int_0^x f_{n-1}(t)dt}$$ The question is, what is $\lim_{n\to\infty} f_n(x)$? The fix ...
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1answer
86 views

A question on Vitali convergence theorem

Let $(X,\mu)$ be a measure space. Vitali convergence theorem says that if (a) $\mu(X)\lt \infty$ (b) $\{ f_n \}$ is uniformly integrable (c) $f_n \to f$ a.e. (d) $|f(x)| \lt \infty $ a.e. then ...
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1answer
163 views

$L_2$ is of first category in $L_1$ (Rudin Excercise 2.4b) [duplicate]

We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$. But in (b) they ask to ...
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1answer
62 views

Differentiability of the function $f(z)=|z|^p z$($p>0$).

Suppose $p>0$ in a real number, is the function $f (z) =|z|^p z$ a differentiable function? Moreover, if $f (z) =|z|^p z$ is differentiable, dose $f$ belong to the space $C^{\left\lfloor p ...
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2answers
143 views

If $D$ is dense in $X$, and $Y\subset X$, what conditions on $Y$ ensure that $D\cap Y$ is dense in $Y$?

Suppose that $(X,\tau)$ is a topological space, $D\subset X$ is dense in $X$ and $Y\subset X$. It can be shown that if $Y$ is open, then $D\cap Y$ is dense in $Y$ (using the subset topology). However, ...
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0answers
113 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
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1answer
570 views

partial derivatives continuous $\implies$ differentiability in Euclidean space

I am given this theorem: If $f \in C^1(A,\mathbb R^m)$, i.e. every partial derivative of $f$ is continuous on $A$, and $A$ is open in $\mathbb R^n$, then $f$ is differentiable on $A$. Is the ...
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1answer
61 views

Question about notation in differential equations.

In general, an ordinary differential equation is in the form $$ \begin{cases} x'(t) = f(t, x(t)) \\ x(t_0) = x_0 \end{cases}. $$ When proving the existence and uniqueness theorems, an operator $T$ was ...
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79 views

Evaluation of the integral.

$$I\left(n,\epsilon\right)=\int_{-{\rm i}\infty}^{+{\rm i}\infty} \frac{{\rm e}^{\epsilon z}}{\left(z+\epsilon\right)^n}\,{\rm d}z$$ The integration is taken along the imaginary axis, an integer ...
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1answer
131 views

contraction point?

This is an interesting question I saw in a book online: Suppose that $g:\mathbb R \to $$\mathbb R$ is a contraction. Then $g$ has a unique fixed point $c$ and that for any number $x_0$, the sequence ...
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2answers
64 views

$A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$.

Construct a sequence of measurable sets $A_k\subset\mathbb{R}$ such that $\lim\sup A_k=\mathbb{R}$ but $\lambda(A_k)=1$ (Lebesgue measure) for all $k$. My thoughts: Since $\lim\sup ...
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1answer
199 views

Prove that $x$ is any positive real number greater than $0$, $x>0$, then exists $N$ in the natural numbers such that $\frac{1}{N^3}<x$

Prove that $x$ is any positive real number greater than $0$, $x>0$, then exists $N$ in the natural numbers such that $\frac{1}{N^3}<x$ My steps: Well I begin with $N\in\mathbb{N}$ and ...
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3answers
122 views

Please more help me to find the convergence interval and the sum -by using residue theory- of the series.

The sum is that $$\sum_{n=0}^\infty \binom{3n}{2n} x^n$$ First of all, I need to check whether the sum converges or not and if it is convergent, which points? I am using ratio test. $$ ...
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2answers
217 views

An exercise in Rudin's RCA

Would you please give me some help on the following problem? Suppose $1 \leq p \leq \infty$, and $q$ is the exponent conjugate to $p$. Suppose $\mu$ is a positive $\sigma$-finite measure and $g$ is a ...
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417 views

Necessary and sufficient conditions for a polynomial $p$ to satisfy $\|x\|\to\infty\implies p(x)\to\infty$?

I'm looking for a necessary and sufficient conditions (I'm not even sure these exist) for a polynomial $p:\mathbb{R}^n\to\mathbb{R}$ to be "radially unbounded", that is $$\|x\|\to\infty\implies ...
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57 views

Prove C[0, 1] is complete with metric a different metric

would any one tell me whether C[0,1] is complete under the following metric $$ \sup_{t\in [0, T]}e^{-Lt}|x(t)-y(t)| $$ and how to prove the claim I know some reasoning on how to prove C[0, 1] is ...
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1answer
223 views

Complex Analysis - Taylor Series

The question reads as follows: Let the function $f$ be given by $f(z)=\frac{z}{2}+\frac{z}{e^z-1}$ if $z\neq0$ and $f(z)=1$ if $z=0$. Show that $f$ is analytic at $z=0$ and that $f(z)=f(-z)$. ...
4
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1answer
107 views

$f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ nonconstant, continuous, with period $1, \sqrt{2}$, respectively, then $f_1 + f_2$ is not periodic

I've been working on this problem for several hours, but I keep getting stuck. Suppose $f_1, f_2 : \mathbb{R} \rightarrow \mathbb{R}$ periodic with period $1, \sqrt{2}$, respectively, and that each of ...
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1answer
223 views

Why is this function not locally Lipschitz?

I was reading an exercise, and supposedly this function: $$\chi \colon \Bbb R\times\Bbb R\to\Bbb R, \quad \chi (t,x)=3x^{2/3}$$ is not locally Lipschitz (in the second variable). In the notes this ...
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1answer
173 views

the solution of Fredholm´s integral equation

Be $\lambda \in \mathbb{R}$ such that $\left | \lambda \right |> \left \| \kappa \right \|_{\infty }(b-a)$. Prove that the solution $f^*$ of the integral equation of Fredholm $$\lambda f ...
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2answers
121 views

sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$.

Give an example in $\mathbb{R}$ of a sequence of Lebesgue measurable sets $A_k$ such that $A_k\subset[0,1]$, $\lim \lambda(A_k)=1$, but $\lim \inf A_k=\emptyset$. My thoughts: By definition, ...
0
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1answer
58 views

Proving $m\lambda(E_m)\le\sum_{k=1}^{\infty}\lambda(A_k)$

Assume $A_1,A_2,...$ are measurable sets. Let $m\in\mathbb{N}$, and let $E_m$ be the set defined as follows: $x\in E_m\iff$ $x$ is a member of at least $m$ of the sets $A_k$. Prove that $E_m$ is ...
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1answer
90 views

Does the property $a\neq a$ exist somewhere in mathematics?

Whenever I read about properties of the real numbers, I'm always presented the property $a=a$ sometime ago I didn't know why they stated such obvious properties but then, after reading some abstract ...
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1answer
811 views

Why is the matrix norm $||A||_1$ maximum absolute column sum of the matrix.

By definition, we have: $||V||_p=\sqrt[p]{\displaystyle \sum_{i=1}^{n}|v_i|^p}$ $||A||_p=sup\frac{||Ax||_p}{||x||_p}$, $x\not=0$ and if $A$ is finite, we change sup to max. However I don't really ...
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356 views

Optimal assumptions for a theorem of differentiation under the integral sign

Let us consider the following integral: $$I(x)=\int_\Omega f(x, \omega)\, d\omega, $$ where $\Omega$ is a measure space and $f\colon \mathbb{R}\times \Omega \to \mathbb{R}$ is such that $f(x, ...
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4answers
126 views

Two form of derivative $ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Why I can write formula derivative $$ f'(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$$ in this form: $$ f'(x)=\frac{f(x+h)-f(h)}{h}+O(h)$$ I know, that it's easy but unfortunately I forgot.
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2answers
87 views

Prove $\int_0^1|f(t)-g(t)|dt \le (\int_0^1|f(t)-g(t)|^2dt)^{1/2} \le \sup_{t\in[0,1]}|f(t)-g(t)|$

Let $C[0,1]$ be the set of all continuous real-valued functions on $[0,1]$. Let these be 3 metrics on $C$. $p(f,g)=\sup_{t\in[0,1]}|f(t)-g(t)|$ $d(f,g)=(\int_0^1|f(t)-g(t)|^2dt)^{1/2}$ ...
1
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2answers
107 views

How to prove that $x/y$ is continuous in R

$f:R^2$ \{y=0} $\Rightarrow R$ , $f:(x,y)\Rightarrow x/y$. Prove (formally) that $f$ is continuous. I think what I should show is that any point that belongs to an open ball of radius $\epsilon$ of ...
0
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1answer
33 views

Asymptotic behaviour of real sequences

Let's say we have two real sequences $(a_n)_{n\in\mathbb{N}}$ and $(c_n)_{n\in\mathbb{N}}$ with $c_n\in o(\frac1n)$ (i.e. $c_n(\frac1n)^{-1}\xrightarrow{n\rightarrow\infty}0$). And for all ...
1
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1answer
51 views

Derivation of weak form of Euler Lagrange Equation

In Giaquinta's and Giusti's 1982 paper entitled "On the regularity of the minima of variational integrals", they look at the following quadratic functional: \begin{equation} ...
5
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1answer
420 views

Set of discontinuous points

Suppose $f$ is function from $\mathbb{R}$ to $\mathbb{R}$. Let be the set $\mathbf{A}$ that contains all the discontinuous points of $f$.Is $\mathbf{A}$ Borel Measureable?
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1answer
63 views

Question on Discrete metric space

Let $X = \{1,1/2,1/4,...,1/2^n,...\} \cup \{0\}$ and $Y = \{X\} - \{0\}$. Is $Y$ dense in $X$? The metric is the usual. If yes, why a separable discrete metric space is then countable? In this ...