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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3
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1answer
384 views

Construction of a sequence of simple functions converging pointwise to a given function

Q1: How to construct a sequence of $\{f_n\}$ of simple function for a function $f$ such that $f_n\to f$ converges pointwise? Q2: If $f$ is measurable is $f_n$ also measurable for each $n$? Q2 is by ...
0
votes
1answer
147 views

Union of countable convergent subsequences

I am trying to come up with example of sequence whose countably many subsequences converge to single point but the original sequences does not. I came up with a series, which seems to converge but ...
5
votes
1answer
199 views

Taylor's Theorem Application Question, $f(x)$ smooth and $f(0)=0$ implies $f(x)/x$ smooth.

I am wondering the following fact, and I believe I know the answer, but I am not sure why. If $f(x)$ is a smooth function from $\mathbb{R}$ to $\mathbb{R}$, if $f(0)=0$, is it true that $f(x)/x$ is ...
1
vote
0answers
69 views

$E=A\cup B$ measurable $\Rightarrow$ $A,\ B$ are measurable. [duplicate]

Possible Duplicate: Proving sets are measurable The problem is: Suppose $E\subset \mathbb{R}$ is (Lebesgue) measurable with $|E|<\infty$, and $$E=A\cup B, \hspace{1cm} A\cap ...
3
votes
1answer
310 views

Is the Sobolev Space $H^k(0,1)$ a banach algebra?

In Adams'book:Sobolev Spaces, I know that if $kp>n,\Omega\subset R^n$ is boundary domain and has cone property, then $W^{k,p}(\Omega)$ could see as a banach algebra. My question is that does it ...
1
vote
3answers
780 views

Open and closed mapping are not necessarily continuous

In the link http://en.wikipedia.org/wiki/Open_and_closed_maps, it says "To every point on the unit circle we can associate the angle of the positive x-axis with the ray connecting the point with ...
2
votes
1answer
312 views

how prove this integral inequality?

How prove that for all continuous and decreasing function $f:[0 ,1]\mapsto(0,+\infty)$ $$\frac{\int_{0}^1x(f(x))^2dx}{\int_{0}^1xf(x)dx}\leq \frac{\int_{0}^1(f(x))^2dx}{\int_{0}^1f(x)dx}$$ thanks in ...
0
votes
2answers
339 views

Vector field with bounded integral curves

I am thinking about smooth vector fields on some (open set of an) euclidean space $\mathbb{R}^n$. I know that the integral curves of a general vector field $X$ are not defined for every time $t\in ...
1
vote
0answers
73 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
3
votes
0answers
155 views

Simplifying an integral arising in Physical Chemistry

I am struggling to understand the following transition (encountered in a paper on Physical Chemistry). Let $$D=\frac{\tau_0^{-1}\int_0^\infty G(t)dt}{1-\tau_0^{-1}\int_0^\infty G(t)\int ...
2
votes
3answers
346 views

For every $f\in C[0,1]$ there is a sequence of even polynomials which converges uniformly on $[0,1]$ to f

For every $f\in C[0,1]$ there is a sequence of even polynomials which converges uniformly on $[0,1]$ to f ? What I have tried: f is continuous on $D:=[0,1]$, let $(x_k)_{k\in \mathbb{N}} \in D$ ...
3
votes
2answers
3k views

What does $C[0,1]$ mean?

In the context of real analysis, I have found this question: For each $$f \in C[0,1] $$ there is a series of even polynomials , which converge uniformly on $[0,1]$ to f. What is $C[0,1]$ ? Is it ...
0
votes
2answers
132 views

A proof to the theorem: $\{s_n\}_n\subset\mathbb{R}$ is convergent if it is cauchy sequence.

I wonder if my idea of proof is correct. Prove that $\{s_n\}_n\subset\mathbb{R}$ is convergent if it is cauchy sequence. idea of pf: By definition, for every $\epsilon> 0\exists N$ ...
2
votes
0answers
150 views

Proving metric( the triangle inequality) [duplicate]

In complex plane $\mathbb{C}$, how to prove that $$d(z_1,z_2)=\frac{|z_2-z_1|}{\sqrt{1+|z_1|^2}\sqrt{1+|z_2|^2}}$$ is a metric? I got stuck in the triangle inequality, and have no idea of proving it, ...
1
vote
1answer
160 views

Jensen's inequality and a estimate in $L^p$

In problem 3 we have: If $f:\mathbb{R} \longrightarrow\mathbb{R}$ is mensurable, $E:=\mathrm{supp}\ f$ and $$\int_E e^{|f(x)|}dx =1,$$ then $f\in L^p(\mathbb{R})$, for all $p\in(0,\infty)$ and ...
1
vote
1answer
372 views

Intuition behind the convolution of two functions

Suppose $f(x)$ and $g(x)$ are two functions. What is intuition or idea behind the convolution of $f$ and $g$? After taking the convolution we will get a new function. What is the geometric relation ...
7
votes
1answer
471 views

Interior Sphere Condition

Let $\Omega\subset\mathbb{R}^N$ be a bounded open set. We say that $\Omega$ satisfies the interior sphere condition (ISC), if for all $y\in\partial\Omega$ there is $x\in\Omega$ and a open ball ...
2
votes
1answer
135 views

Proof that Laplacian is surjective $\mathcal{P}^n\to\mathcal{P}^{n-2}$

Let $\mathcal{P}^n$ denote the vector space of homogeneous polynomials on $\mathbb{R}^3$ of degree $n$. I need to prove that $\Delta|_{\mathcal{P}^n}:\mathcal{P}_n\to\mathcal{P}_{n-2}$, for $n\geq2$ ...
2
votes
1answer
212 views

A problem about convergence…

I am studying an article of Berestychi-Caffarelli-Niremberg - Monotonicity for elliptic equations in unbounded Lipschitz domains, and I don't understand a convergence in the demonstration of the lemma ...
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0answers
1k views

Proof that sum of continuous functions is continuous in metric space.

Definition: A function $f: X \to Y$ between metric spaces $X$ and $Y$ is continuous in $x \in X$ iff for every $\varepsilon > 0$ there exists a $\delta > 0$ such that for all $x' \in X$ with ...
3
votes
2answers
139 views

Does $d(x+u, y + v) \le d(x, y) + d(u,v)$ holds for every metric?

The title said it, I want to prove that $$ d(x+u, y + v) \le d(x, y) + d(u,v) $$ for every metric $d$. If the metric is induced by a norm, i.e. $d(x,y) := ||x-y||$, then this is easy. \begin{align*} ...
2
votes
2answers
374 views

Homeomorphic and Isometric Spaces

Problem I'm currently studying metric spaces, and the lectruer's notes make the remark: Clearly $(0,1)$, $(0,\infty)$ and $\mathbb{R}$ are homeomorphic under the standard metrics, but no two of them ...
0
votes
1answer
32 views

Information needed about Local Extremas in differential calculus

We know a function $f \in C^2(R)$ has a Local Maximum in the origin $(0,0)$. What can you say about the differential: $d_{(0,0)}^2f(1,-1)<0$? I've recently got this on a test and I'm not sure if ...
4
votes
1answer
320 views

Looking for proof that an open set in vector space contains the sum of two open sets.

Problem: To show that, in a topological vector space, for a given neighborhood of zero $W$, there exist two neighborhoods of zero, $V_1$, $V_2$, whose sum is contained in the first neighborhood, ...
4
votes
0answers
83 views

Quotient-lifting properties

I borrowed this terminology from K. Conrad's article on series of subgroups, in which he discusses solvability of groups. This property of certain groups satisfies Let $N\triangleleft G$. Then ...
0
votes
1answer
379 views

Almost everywhere pointwise convergence of continuous functions

Suppose I have a sequence of continuous functions converging pointwise almost everywhere to a continuous function. Is it true that we have pointwise everywhere convergence?
2
votes
2answers
65 views

Is there some difference between the two definition of integration along curves about complex function

In Stein's Complex Analysis, the integral of $f$ along $\gamma$ is defined by $$\int_{\gamma}f(z)\text{d}z=\int_a^bf(z(t))z'(t)\text{d}t$$ where $z:[a,b]\rightarrow \Bbb{C}$ is a parametrization of ...
0
votes
1answer
56 views

Given a function space with a norm , what is the meaning of writing $||.||$ when the used norm is $||.||_\infty$

Example 1 Given $$C_{0}(\mathbb{R}^{n})=\{f\in C(\mathbb{R^n} \ | \ \ \exists R \ge 0 \ \text{such that } f(x)=0 \ \text{for} \ ||x||\ge R \}$$ and $$||f(x)||_{\infty} = \max_{x\in R^n}|f(x)| $$ ...
1
vote
1answer
80 views

Is $(C_{0}(X), ||.||_{\infty})$ a Banach Algebra?

Is $(C_{0}(X), ||.||_{\infty})$ a Banach Algebra? Given $$C_{0}(\mathbb{R}^{n})=\{f\in C(\mathbb{R^n} \ | \ \ \exists R \ge 0 \ \text{such that } f(x)=0 \ \text{for} \ ||x||\ge R \}$$ and ...
0
votes
2answers
824 views

example of a function with compact support

Can you give an example of a function which is $C^\infty (\mathbb{R})$ having support on (-1,1) such that $ \int_{-1}^1 f(x)\,dx$=1 and $ \int_{-1}^1 xf(x)\,dx$=0. Thank you.
4
votes
2answers
187 views

how to prove this question about derivative and differentiation

Let $$ f:\mathbb{R}\to \mathbb{R} $$ such that $f ',f'',f'''$ exist and $\lim_{x\to+\infty} f(x)=t$ exists if $ \lim_{x\to+\infty} f'''(x)=0$. Then prove that $$ \lim_{x\to+\infty} f'(x) = ...
0
votes
2answers
87 views

Differentiability on vector values function

I just had my first lecture of my analysis course, and we were introduced the differentiation on general euclidean space where the derivative is regarded as a linear transformation. Define ...
3
votes
1answer
147 views

Second order linear ODE with variable coefficients

Consider the second-order linear differential equation $u'' + p(x)u' + q(x)u = 0$ where $p$ and $q$ are continuous on the entire $\mathbb{R}$. Suppose that $q(x) < 0 $ everywhere. Show that if $u$ ...
3
votes
6answers
882 views

How to prove $ {a_n} = \frac{n!}{2^n}$ diverges to $+ \infty$?

I would like to prove that the sequence $ {a_n} = \frac{n!}{2^n}$ diverges to $+ \infty$. As I understand it, this means that for all numbers $M$, I must find a number $N$ such that for all $n \ge N$, ...
0
votes
0answers
43 views

What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $?

What are the features of $n/d, n \rightarrow d, d \rightarrow \infty; n, d \in \mathbb{N} $? What is the value of $\lim_{n \rightarrow d, d\rightarrow \infty} (n/d)$? What is the function's range? ...
11
votes
3answers
251 views

If $\int_a^b f(x) \ \mathrm{d}x = \int_a^b g(x) \ \mathrm{d}x$ then $\exists x \in [a,b]$ with $f(x) = g(x).$

I am trying to prove the following: Take $f, g:[a,b] \to \mathbb{R}$ such that $f$ and $g$ are continuous. If $$\int_a^b f(x) \ \mathrm{d}x = \int_a^b g(x) \ \mathrm{d}x,$$ then there exists some ...
4
votes
1answer
115 views

Hardy space question

Let $T$ be the unit circle. Let $\phi\in C(T)$ and let $\psi$ be a function in $L^2(T)$ such that $\phi+i\psi\in H^2$. Assume both $\psi$ and $\phi$ are real-valued. Show $e^{\phi+i\psi}\in ...
0
votes
1answer
136 views

Weierstrass Approximation— Rudin's Proof

1) In Rudin's proof towrds the end, he declares that $4M\cdot \sqrt{n}\cdot(1-\delta^2)^n$ goes to zero as n goes to infinity. I just don't see it though -- how is this the case. 2) My second ...
2
votes
2answers
291 views

Approximate continous function with linear growth condition by Lipschitz function

Suppose a continuous function $f(u)<K(1+|u|)$ for some positive number $K$. How can we find a sequence of Lipschitz functions $f_{n}$ that converge to $f$ uniformly on $\mathbb{R}$. If we require ...
2
votes
2answers
753 views

how prove GL(n,R) is not connected subset and open subset of$M_n (\mathbb{R})$with this distance

let n>1 be natural and fix number, $S:=${A : $M_n (\mathbb{R})$ be all real matrix,define this meter for all $A=[a_{ij}]$ $B=[b_{ij}]$ d(A,B):=max{|$a_{ij}-b_{ij}$|:i,j=1,2,2...,n} and GL(n,R) is ...
2
votes
1answer
69 views

Continuity of $x+y$ and $xy$ in $\mathbb{R}^{\infty}$

How can I show (or where can I find) that in $\mathbb{R}^\infty$: $f(\textbf{x},\textbf{y})=\textbf{x}+\textbf{y}$, $g(\textbf{x}, k)=k\cdot \textbf{x}$ are continuous functions? ($g$ is from ...
1
vote
1answer
43 views

Dose the series diverge on the boundary when $\theta $ is irrational?

series: $$\sum_{n=1}^{\infty}\left(\ln n\right)^2z^n$$ where $z\in \Bbb{C}$ , by Hadamard's formula, the radius of convergence is $1$, and I try to discover the status of convergence when ...
0
votes
1answer
625 views

Is the space of bounded functions with the Supremum norm a Banach Algebra

X is an arbitrary , non empty set, B(X) the set of bounded functions $f:X\rightarrow \mathbb{R}$ and $||f||_\infty = \sup_{x\in X }|f(x)|$. Is $(B(X),||.||_\infty )$ a Banach Algebra? My attempt ...
3
votes
2answers
147 views

Asymptotic rate of growth of a sum

Consider $$\Phi_0(x) = \sum_{i=0}^{\infty} (1-x)^i,$$ where $x \in (0,1)$. As $x \rightarrow 0$, $\Phi_0(x)$ blows up as $\Theta(1/x)$. Similarly, consider $$ \Phi_1(x) = \sum_{i=0}^{\infty} i ...
2
votes
2answers
397 views

Upper bound for the absolute value of an inner product

I am trying to prove the inequality $$ \left|\sum\limits_{i=1}^n a_{i}x_{i} \right| \leq \frac{1}{2}(x_{(n)} - x_{(1)}) \sum\limits_{i=1}^n \left| a_{i} \right| \>,$$ where $x_{(n)} = \max_i x_i$ ...
6
votes
1answer
243 views

Is the following function continuous at $x = 0$?

Define $f: \mathbb{R} \to \mathbb{R}$ by $$ f(x) = \cases{ x - 1 \ \ \text{ if } x \in \mathbb{Q} \\1 - x \ \ \text{ if } x \not\in \mathbb{Q}. }$$ I'm trying to prove whether or not $f$ is ...
1
vote
1answer
70 views

Let $I=\int_{0}^{1}f(x)x^2dx$

I came across the following problem that says: Suppose $f$ is a continuous real-valued function. Let $I=\int_0^1 f(x)x^2 \, dx$.Then it is necessarily true that $I$ equals : ...
0
votes
1answer
512 views

Is the Space of bounded functions with the maximums norm a Banach space and even a Banach Algebra?

X is a arbitrary non empty set , B(X) the set of bounded functions $f:X\rightarrow \mathbb{R}$ and $||f||_\infty = \sup_{x\in X} |f(x)|$ Completeness: Let $(f_n(x))_{n \in \mathbb{N}}$ be a cauchy ...
0
votes
2answers
245 views

Prove that there is a $\delta$ such that $\int_{0}^{1} (f(x))^2dx\leq \delta$$\int_{0}^{1} (f'(x))^2dx$ for all $f$ with these conditions

Let $S=\{f:\mathbb{R} \to \mathbb{R}\}$ that satisfies: $\forall f\in S$, $f'$ exists and $f'$ is continuous and $f(0)=f(1)=0$. Please prove that $\exists \delta :\forall f\in S$ s.t. $\int_{0}^{1} ...
2
votes
1answer
237 views

Determine the range of $f(x) = \log\frac{1+\sin(x)}{1-\sin(x)}$

Determine the range of $f(x) = \log\frac{1+\sin(x)}{1-\sin(x)}$ and the preimage $f^{-1}([0, \log3>)$. For the Range, I divided the function into a composition of 3 functions where $$f_1 = ...