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
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1answer
172 views

Open Cover for a Compact Subset

I am doing some extra exercises for an Analysis class, and I found this one. We haven't seen much of what an open cover is, but I want to learn it. So, here it goes, and thank you everyone! Let ...
1
vote
1answer
46 views

Coefficients of Chebychev Polynomials

Is there a known formula for the coefficient of x^k in the nth chebychev polynomial of the first kind?
1
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1answer
124 views

Show: $C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)$.

Show that $$ C^1(\Omega)\subset C^{0,1}(\Omega)\subset C^{0,\lambda}(\Omega)\subset C^0(\Omega)~~~~~~~\forall0<\lambda\leq 1. $$ Hello, I have some problems to show these ...
2
votes
1answer
45 views

preimages of simple functions form a partition

Let $\varphi $ ba simple, then we know $A_i = \varphi^{-1} (a_i) $ . Claim is $A_i$ paritition $\mathbb{R}$ my try: Note that the sets $A_i = \varphi^{-1} ( \{ a_i \} ) $ form a partition of ...
2
votes
1answer
161 views

Is space of measures Inner product space

Let $(X, \mathcal{F})$ be measurable space and let $\mathcal{M}$ be space of all signed measures on it. It is clear that, $\mathcal{M}$ is a Real vector space. I am interested to know if there is any ...
0
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2answers
135 views

Polynomial convergence to zero

Let $f_k $ be a series of n degree polynomials, that converges to $0$ uniformly in $[-M, M] $ for every $M$. Say $f_k = a_{(0,k)} + a_{(1,k)}x +... + a_{(n, k)}x^n$ Prove that for every i, $a_{(i, ...
0
votes
1answer
87 views

About Folland prop 2.7

Proposition 2.7 in Folland emphasizes that $f: X\rightarrow \bar{\mathbb{R}}$. Does the same conclusion hold for $f:X\rightarrow \mathbb{R}$? Proposition 2.7 Suppose $f_n:X\rightarrow ...
2
votes
1answer
229 views

Reflexivity of a Banach space without the James map

The reflexivity of a Banach space is usually defined as having to be enforced by a particular isometric isomorphism. Namely the map that takes each element to the evaluation, which is already an ...
1
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1answer
89 views

Markov inequality limit

Let $f: \mathbf{R}^d \rightarrow [0,\infty]$ be a Lebesgue integrable function. Prove $$\lim_{\alpha \rightarrow \infty} \alpha m(\{x:f(x)>\alpha\})=0.$$ Hint: For $\epsilon>0$ take $g$ simple ...
3
votes
2answers
71 views

Proof of $|x^{\alpha} - y^{\alpha}| \le \alpha^{\alpha} |x-y|$ for $\alpha \ge 1, x,y\in [0,1]$

I want to prove $$ |x^{\alpha} - y^{\alpha}| \le \alpha^{\alpha} |x-y| $$ for $\alpha \ge 1$ and $x,y \in [0,1]$. For $\alpha \in \mathbb N$ I already got the proof by using the formulae $$ (x^n - ...
2
votes
1answer
489 views

sup of A Union B

Assumgin A,B, two set are upper-bounded. I need to prove that A Union B is also upper bounded and the supremum is max(supA, supB). This question can be explained intuitivly, but how do you prove it ...
2
votes
1answer
130 views

How Find this $f(x,y)$ such $f''_{xy}$ is not exsit.but $f_{xx}'',f_{yy}''$ is exsit.

Question: take example:such $f(x,y)$ in the unit disc $ D$ that extends continuously to ∂D such that $f_{xx}'',f''_{yy}$ exist,and continuously,But $f''_{xy}(0,0)$ is not exsit. My try: I ...
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0answers
52 views

Property of Laplace transforms

I was looking at this answer to the question asked and I am curious about the $$\int_0^\infty F(u)g(u) du = \int_0^\infty f(u)G(u) du $$ relationship being used. I referred to the link provided in ...
0
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1answer
32 views

Prove that $Tu(x)$ is a contraction. $Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$

I want to show that $Tu(x)$ is a contraction where $$Tu(x) = -\lambda\int_0^1g(x,y)\sin(u(y))\,dy$$ and $$g(x,y) = \begin{cases} x(1-y) & 0\leq x\leq y\leq 1, \\ y(1-x) & 0\leq y \leq x \leq ...
3
votes
1answer
49 views

$x^2 f^{''}(x)+4xf^{'}(x)+2f(x)\geq 0$, prove $f(x)\leq 0$

More specifically, suppose $f$ is continuous on $[a,b]$ with $f(a)=f(b)=0$ and $x^2f^{\prime \prime}(x)+4xf^{\prime}(x)+2f(x)\geq 0$ for $x\in (a,b)$. Prove that $f(x)\leq 0$ for $x\in [a,b]$. I'm ...
3
votes
1answer
160 views

Classifying non-unique solutions to ODEs

The canonical example of an ODE with nonunique solutions is $y'=\sqrt{y}$ with $y(0)=0$. The solutions are $y=\frac{1}{4}t^2$ and $y=0$. We also know that just by looking at $\sqrt{y}$, the fact that ...
2
votes
1answer
167 views

Volume of ellipsoid

How do I calculate the volume of the intersection of ellipsoid $x^2/36+y^2/49+z^2/49\leq 1$ and subspace $x/6+y/7+z/7\leq1$ ?
1
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1answer
71 views

Example of two norms and ONE linear operator that is bounded and unbounded in a norm.

I am looking for an example of a linear operator that is bounded as well as unbounded depending on which norm you take. Since I do not have much experience with Functional Analysis, I do not know many ...
1
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1answer
43 views

Show that all numbers can be written as $x=k\alpha+x'$, given…

The question is: Suppose $\alpha>0$. Prove that any real number $x$ can be written uniquely in the form $x=k\alpha+x'$, where $k$ is an integer and $0\leq x'\lt\alpha$. How do I approach this ...
1
vote
1answer
47 views

Integration of $\int_{x(0)}^{x(t)}\frac{1}{\sqrt{|y|} }dy$ for $y<0$

I tried to solve $x'(t)=\sqrt{|x(t)|}$ by using separation of variables. So I did $$\int_{0}^{t}\frac{x'(s)}{\sqrt{|x(s)|}}ds$$ and used the substituion $y=x(s)$, which gave me the integral ...
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0answers
43 views

Generator of a transport semigroup on the torus

Consider on the Banach space $C( \mathbb T;\mathbb R)$ of continuous functions on the 1-dimensional torus $\mathbb T:=\mathbb R/\mathbb Z$ (equipped with uniform norm) the operator $L$ given by $$ L ...
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0answers
134 views

Questions about the Gateaux derivative

We defined that a function is Gateaux differentiable, if all directional derivatives exist. I just wanted to check, whether I got a few things right: Now I wanted to ask, whether it is true that if ...
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2answers
143 views

$\dfrac{\sin x}{x}$ modified improper integrals.

I am trying to evaluate this integrals: $$ \int_{-\infty}^{\infty} \! \left[\frac{\sin\left(x\right)}{x}\right]^n \, \mathrm{d}x. $$ I know how to prove it if $n=1$ using Fourier Transform, but I ...
2
votes
0answers
288 views

Difference of two convex functions

This is an exercise from a probability textbook on Ito's formula, basically Ito's formula extends to functions of this type. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f$ is ...
2
votes
1answer
114 views

Theorems related to the Sturm-Picone comparison theorem.

I was reading in some course notes on differential equations last year from a URL I can no longer find, and there was a problem that has itched me because I always felt like there was an error in the ...
1
vote
2answers
309 views

Does the set of rational numbers between 0 and 2 have the least upper bound property?

Let $A = \{ a \in Q : 0 < a < 2\}$ Does A have the least upper bound property? Definition: $A$ has the least upper bound property if $\forall$ nonempty $B \subseteq A$, if $B$ has an upper ...
3
votes
3answers
77 views

Analysis question on Integration bounds

I have to find all functions f(t) such that $\int_x^{x^2} f(t) d t=\int_1^x f(t) dt$ I think the solutions are all the functions of the form a/(x+b) because the logarithm would divide the two powers ...
0
votes
1answer
82 views

log partition function of exponential family

In an exponential family $$p_{\theta}(x)=\exp \left(h(x)+\sum\limits_{i=1}^s \theta_iT_i(x) - \phi(\theta) \right) $$ is the log partition function $$ \phi(\theta)=\log \int \exp ...
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0answers
40 views

Homologously of connected components

How can I prove that if $\Omega$ is homologously connected $\Rightarrow$ every connected component of $\Omega$ is homologously connected?
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0answers
51 views

Continuity of functions between Metric Spaces [closed]

Prove that every function mapping $(X,d_0)$ to a metric space $(Y,d)$ is continuous. where $d_0$ is the discrete metric.How do we prove this?
0
votes
1answer
56 views

Uniform convergence and cauchy sequence

if I have a sequence $(x_n) \subset (C[0,1],||.||_{\text{max}})$ such that $sup_{s \neq t} \frac{|(x_n(t)-x_m(t)) - (x_n(s)-x_m(s))|}{|s - t |},s,t \in [0,1] $ is convergent to zero. And we have that ...
0
votes
1answer
203 views

The product of countable spaces which have countable dense subset has a countable dense subset.

Let $X$ be a product of countably many spaces $X_{i}$ which have countable dense subset. Prove that $X$ has a countable dense subset.
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1answer
29 views

Can we construct two sets and functions for the given conditions?

Can we construct two sets $A$, $B$ and two invertible functions (one to one) $f_A \in \mathbb{R}^n$, $f_B\in \mathbb{R}^n$ such that the following conditions are satisfied? The conditions are ...
0
votes
1answer
52 views

prove that $X \times Y$ is a Lindelöf space

Let be $X$ a Lindelöf space and $Y$ a compact space, prove that $X \times Y$ is a Lindelöf space.
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3answers
160 views

Show that $f(X)$ has a dense subset.

Let $X$ be a topological space which has a dense countable subset $D$, and suppose $f\colon X \to Y$ a continuous function. Show that $f(D)$ is dense countable in $f(X)$.
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3answers
116 views

Find $\lim_{n\to\infty} \frac{n^2}{2^n}$

$$\lim_{n\to\infty}\frac{n^2}{2^n}$$ Do you have some tips so I could solve this problem, without the use of L'Hôpital's rule? Indeed, we didn't see formally L'Hôpital's rule, nor Taylor series so ...
1
vote
3answers
329 views

If $\lim\limits_{x\to\infty} xf(x) = L$, then $\lim\limits_{x\to\infty} f(x) =0$ [duplicate]

Show that if $f: (a,\infty) \rightarrow \mathbb R$ such that $$\lim_{x\to \infty} xf(x) = L$$ where $L \in \mathbb R, $ then $$ \lim_{x\to \infty} f(x) = 0. $$
9
votes
1answer
160 views

Complete ordered field is an Archimedean field that cannot be extended to an Archimedean field

As a bonus problem, our professor of real analysis asked us to prove that the real numbers (a complete ordered field) cannot be extended into an Archimedean field, with no definition of what he meant ...
2
votes
3answers
92 views

A simple polynomial differential equation

Consider the differential equation $$ \frac{d}{dx}y(x) = -(y(x))^3. $$ with initial condition $y(0) = 1$. I know that it admits the unique solution $$ y(x) = \sqrt{ \frac{ 1 }{ 1 + 2 x } }. $$ ...
0
votes
1answer
93 views

Find all $f: \mathbb{Q} \rightarrow \mathbb{R}$ such that $f(x+y) = f(x)+f(y)$ [duplicate]

i have to find all functions $f: \mathbb{Q} \rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$. So functions of the form $f(x) := ax, a \in \mathbb{R}$ satisfy the above condition: $$ ...
0
votes
1answer
147 views

Uniform convergent and lipschitz continuous

I want to prove that if I have a sequence $ f_n\in C[0,1]$ that is uniform convergent to zero and all functions are lipschitz continuous, then the lipschitz constants form a zero sequence. Does ...
1
vote
1answer
70 views

Does $\int_{0}^{\pi/2n}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t\leq \pi$ hold for $n \geq 2$?

Today I am trying to prove an integral inequality: $$\frac{1}{\pi}\int_{0}^{\pi/2}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t<\frac{2+\ln n }{2}$$ where $n\geq 2$ and $n \in \Bbb{N}$. First, ...
2
votes
0answers
86 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
6
votes
4answers
98 views

How to prove $\sum_{k=1}^n \frac{2^k}{k}< 3\frac{2^n}{n}$?

How to prove $$\sum_{k=1}^n \frac{2^k}{k}< 3\frac{2^n}{n}$$ and further $$\lim_{n\rightarrow \infty}\frac{n}{2^n}\sum_{k=1}^n \frac{2^{k}}{k} = 2$$? These results are verified by computer, yet I ...
1
vote
1answer
24 views

about restricting outer measures

If $\mu_0$ is an outer measure on an algebra, we can extend the premeasure to an outer measure $\mu^*$. By Caratheodory's theorem, the collection of $\mu^*$-measurable sets is a $\sigma$-algebra. Is ...
1
vote
2answers
221 views

Use mathematical induction to prove that for any $k \in\mathbb N , \lim (1+k/n)^n = e^k$.

Use mathematical induction to prove that for any $k \in \mathbb N, \lim (1+k/n)^n = e^k$. I already used monotone Convergence Theorem to prove $k=1$ case. Do I just need to go through the same ...
7
votes
4answers
2k views

What is the relationship between Fourier transformation and Fourier series?

Is there any connection between Fourier transformation of a function and its Fourier series of the function? I only know the formula to find Fourier transformation and to find Fourier coefficients to ...
0
votes
2answers
65 views

Real Analysis: Covergence Question

Suppose that $( x_n )$ is a sequence of real numbers, $( y_n )$ is a bounded sequence of non-zero real numbers, and that $\lim x_n/ y_n = 1$. Prove that $\lim (x_n - y_n) = 0$. Since $y_n$ is bounded, ...
4
votes
0answers
127 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
2
votes
3answers
103 views

Value of tan(pi/2)

I understand that this is a very stupid question but I'm not getting the answer. At x=pi/2,what s the value of tan(x)?Should it be -infinity or +infinity? Texts tell it to be +infinity.But why? ...