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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70 views

modern analysis: limits, integrals, uniform

Suppose $\{f_n\} \to f$ uniformly on $[a, b]$ and both $f$ and the $f_n$ are integrable. Prove that $\lim_{n\to \infty}\int_{a}^bf_n(x)dx = \int_{a}^bf(x)dx$
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1answer
73 views

Signed measure defined by an integral

Let $ (X,\mathcal{M},\mu)$ be a measure space and let $ f:X\to[-\infty,+\infty]$ be an integrable function (i.e. at least one of $ f_+ $ and $ f_-$ is integrable). I want to prove that $ ...
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1answer
61 views

Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
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2answers
70 views

Supremum, and an increasing function

let $f(x)$ be an incresing function, and let $C$ be a constant s.t. $f(x) \leq C.$ Put $D = $ sup $f(x)$, and i need to show that $f(x) \rightarrow D$ as $x \rightarrow \infty$. It seems 'obvious' ...
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1answer
96 views

Linear functional on the set of bounded functions

Let $S$ is non-empty set, set $$l^\infty(S)=\{f:S\rightarrow\mathbb{R}: \|f\|_\infty =:\sup_{x\in S} |f(x)|<\infty\}.$$ Suppose that $\psi:l^\infty(S)\rightarrow\mathbb{R}$ is a bounded linear ...
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1answer
40 views

Prove $g$ is continous on metric space.

For:$(X,d)$ is a metric space , $f:X\to X$ is a continous function and $g:X\to R, x\mapsto g(x)=d(f(x),x)$. Prove that $g$ is a continous function. Definition: $f$ is continous at $x_0$ if only if ...
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2answers
125 views

For which $\alpha$ does the series $\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}}\!\! - \!1\big)$ converge?

For what $\alpha$, does $\displaystyle\sum_{n = 1}^{\infty}\big(2^{n^{-\alpha}} - 1\big)$ converge? Divergence of $\sum_{n = 1}^{\infty}(2^\frac{1}{n} - 1)$ prompted this question.
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1answer
59 views

Sequence of functions whose integrals converge to $0$.

I encountered the next problem and I'm having problems solving it. It says: Let $(X,\mathcal{S},\mu)$ be a measure space and let $\{A_n\}$ be a sequence of measurable sets satisfying $0< ...
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1answer
70 views

Sum of a certain series related to the primes

It is well known that $$\sum_{n > 0}\frac{1}{n}$$ diverges, but $$\sum_{n > 0}\frac{1}{n^2} = \frac{\pi^2}{6}$$ converges. Similarly, $$\sum_{p}\frac{1}{p}$$ diverges, but $$\sum_{p} ...
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1answer
88 views

Showing that $\log \log(z)$ is Analytic (Proof Verification)

Goal: Convert $\log \log (z)$ into a single-valued function defined on a suitable region of $\mathbb{C}$ and then prove that it is analytic. Attempt: As has been demonstrated elsewhere, we have ...
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1answer
79 views

Weak convergence in a subspace

Let $V$ be a normed linear space and $W$ a closed subspace of $V$. Suppose a sequence $\{w_{n}\} \subset W$ and $w \in W$ with $w_{n}$ converges to $w$ weakly in $V$. Why must $w_{n}$ converge weakly ...
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1answer
28 views

why a lemma shows well-definedness of linear transformations

The following lemma can be used to show that some linear transformations are well-defined. I don't quite see that. I mean, if a linear transformation $T$ is well-defined, then if $x=y$ then ...
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5answers
270 views

Does the series $\,\displaystyle\sum_{n = 1}^{\infty}\left(2^{1/n} - 1\right)\,$ converge?

I'm trying to determine if the following sum converges or diverges (this is question 38 in section 11.7 of Stewart's Early Transcendentals): $$\sum_{n = 1}^{\infty}(2^{1/n} - 1)$$ I've considered ...
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2answers
69 views

Is $f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$ bounded?

$$f(x)=\sum_{k\in\mathbb N}\frac1k\sin\frac x{2^k}$$Is this function bounded? So obviously this converges because $|\frac1k\sin\frac x{2^k}|<|\frac x{2^k}|$ and $\sum\frac x{2^k}$ converges by ...
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1answer
41 views

Does $C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})=C^1([0,T];H^{s+1})$?

I have asked this question with other problems, but about this part nobody answers. So I want to ask again, and put some details in it. My question is whether the following equality is correct. If it ...
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2answers
55 views

Sequence Convergence Confusion

$ \lim_{n\to \infty} \frac{3+4n^2}{2n^2 - n } = 2$ We must use the definition of a convergent sequence: "A sequence $(s_n)$ is said to converge to the real number $s$ provided that for every ...
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0answers
308 views

Showing that $\sqrt{1+z} + \sqrt{1-z}$ is Analytic (Proof Verification)

Ahlfors: Give a precise definition of a single-valued branch of the function $\sqrt{1+z} + \sqrt{1-z}$ in a suitable region, and prove that it is analytic. Is my following proof attempt valid? ...
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1answer
884 views

L1 convergence gives pointwise convergent subsequence

I have been reading Terry Tao's notes on Real Analysis and there's a part he just says, but does not really explain, so I am wondering if someone here would. The notes are ...
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1answer
39 views

Sequence does not converge weakly in $\ell^{\infty}(\mathbb{N})$

Let $V = c_{0}(\mathbb{N})$, the space of sequences which converge to 0. Then $V^{\ast} \cong \ell^{1}(\mathbb{N})$ and $V^{\ast\ast} = \ell^{\infty}(\mathbb{N})$. Let $a_{n} = (0, 0, \ldots, 0, 1, 1, ...
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4answers
161 views

Solving the integral $\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$

I really don't know how to solve this integral $$\int_0^{\pi/2} \frac{\sin((2n+1)t)}{\sin t} \mathrm{d}t$$ Should I use firstly a formula of $\sin(a+b)$?
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1answer
60 views

Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
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1answer
73 views

Does Taylor's theorem apply here?

Let $U\subset \mathbb{R}^n$ be open and $f:U\to \mathbb{R}^n$ with $x\in U$ and $\xi$ sufficiently small. Suppose that the following hold: $f(x+\xi)=\sum_{\alpha=0}^k ...
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0answers
54 views

Measures and Outer Measures

I've a "little" doubt. From Carathéodory-Hahn Extension Theorem, we know that, starting from a measure space $\ (X,\mathcal{M},\mu)$, we are always able to obtain an outer measure $\ \mu ^* $. Then ...
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2answers
210 views

Verify the limit using the definition of convergence of a sequence

I know that the definition is: "A sequence $(s_n)$ is said to converge to the real number $s$ provided that for every $\epsilon > 0$ there exists a natural number $N$ such that for all $n \in ...
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0answers
36 views

Two different ways to generate the topology of convergence in measure

Consider the measure space $(X, \mathcal{B}, \mu)$ where $\mu(X) < \infty$. Let $L(X)$ denote the space of measurable functions on $X \rightarrow \mathbb{C}$. Then one way to define the topology ...
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0answers
80 views

Relationship between weak and strong closure

Let $E \subset X$ be a convex subset of a normed space. I want to show that the weak closure of $E$ denoted by $\overline{E}_{w}$ is the same as the strong closure of $E$ denoted by $\overline{E}$. By ...
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0answers
54 views

When integration by parts applies?

I think integration by parts is really useful, but I don't know when integration by parts applies. Take the following proposition as an example: Let $\Omega \subset \mathbb{R}^n$ be an open set, ...
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1answer
36 views

Can someone explain Sensitivity Analysis to a computer programmer

There are lots of long texts about Sensitivity Analysis but as a programmer I get easily bored or can't understand it. Can someone briefly explain Sensitivity Analysis to a programmer? Thanks.
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2answers
113 views

Does $dx$ in the formula $\int f(x)dx$ represents a differential of x?

While I asked a question about integrals Is $dxdy$ really a multiplication of $dx$ and $dy$?, I found out that many of the answers were assuming that dx is just a notation in the formula $\int ...
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0answers
52 views

Taylor series like polynomials

Let $U$ be an open subset of $R^n$ and $f:U\rightarrow \mathbb{R}$ a function and $x\in U$ such that in a small neighbourhood of $x$ and for $\epsilon \in \mathbb{R^b}$ sufficiently small we have the ...
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1answer
218 views

Homeomorphism on the unit circle

Can somebody tell me how to prove that $f:[0,2π)→S^1$ given by $t↦⟨\cos t,\sin t⟩$, where $S^1$ is the unit circle in the plane, and $[0,2π)$ is the real interval, 1. is continuous at the point 0 (the ...
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1answer
185 views

Showing that the space of Hilbert-Schmidt operators form a Banach space.

How do i show that the set of Hilbert-Schmidt operators $HS(H) = \{T \in B(H) \; : \; \sum^{\infty}_{n=1}\|Te_n\|^2 < \infty \}$ for some countable ONB $\{e_n\}$, on a separable Hilbert Space ...
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1answer
35 views

understanding a proof involving equivalence of norms in finite dim. linear normed spaces

I am reading the proof of the theorem shown below (from Linear Functional Analysis by Rynne and Youngson). I can't figure out why the part I highlighted in red is true. I understand why $S$ is compact ...
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1answer
66 views

Banach space problem

I came across the following problem: For an open set $U$ in $\mathbb{R^n}$ we define the set of all k-times continuously differentiable functions $f:U\rightarrow \mathbb{R}$ for which $D^\alpha f$ ...
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0answers
56 views

How to give an epsilon-delta proof of this limit statement? [duplicate]

Although I know a couple of proofs of the statement $$ \lim_{x \to 0 } \frac{\sin x}{x} = 1, $$ I would like to be able to come up with a proof using the definition of the limit (i.e. an ...
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1answer
145 views

Disproving the mean-value theorem of calculus to complex functions?

I'm defining a function $f(z) = z^3 + 1$, and I will let two points $$z_1 = \frac{-1+i\sqrt3}{2}\quad\text{and}\quad z_2 =\frac{-1-i\sqrt3}{2}.$$ I am trying to show that there is no point $w$ on ...
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1answer
44 views

Integral inequality in $\Bbb R^n$

I came across this problem : Let $f\colon [a,b]\rightarrow \mathbb{R}^n$ a continuous vector valued function. Then it is true that: $$\left\Vert\int \limits_a ^b f(t) dt\right\Vert \leq \int ...
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1answer
135 views

Bounded Jacobian implies uniform continuity

I am trying to solve the following problems but I am not sure what the difference between the 2 problems is. 1) Prove that is $U = B_r(x)$ (open ball centered at $x$ with radius $r>0$) is an open ...
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1answer
36 views

Weber function takes on real values?

Why does the Weber function take on real values? In the middle of the proof, the author argues... "cleary it is real-valued." I don't follow the argument. Info about the Weber function ...
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1answer
29 views

Automorphism and endomorphism of the Toeplitz algebras

Let $\ H=l^2(\mathbb{Z}_+)$ be a Hilbert space with orthonormal basis $\ {e_k}$, and $T$ will be right shift operator, t.i. $Te_k=e_{k+1}$. $C^*$ algebra generated by T is a Toeplitz algebra and ...
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1answer
68 views

Product of weakly convergent sequence and sequence boundedly convergent in measure

Question: Let $\Omega \subset \mathbb{R}^d$ be open and bounded, $f, f_n \in L^2 (\Omega)$ and $f_n \rightarrow f$ boundedly in measure (meaning that $f_n \rightarrow f$ in measure and $sup\ ...
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1answer
79 views

A Lebesgue measurable universal Borel function

In 1918 Sierpiński constructed a Lebesgue measurable real-valued function on $[0,1]$ which isn't bounded above by any Borel function (I couldn't find the original reference, but here is a pdf of a ...
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1answer
44 views

About the space $u \in C([0,T];H^{s+1}) \cap C^1([0,T];H^{s})$

I am reading Taylor's Partial differential equations III (nonlinear equations) (Section 1 of Chapter 16, Theorem 1.2), and Sogge's Lectures on Non-linear wave equatuions. I notice that in the energy ...
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2answers
1k views

Prove $1-1/2-1/3+1/4+1/5-1/6-1/7+\cdots$ converges.

Consider the series: $1-1/2-1/3+1/4+1/5-1/6-1/7+\cdots$ Is it convergent? I believe I need to find a way to split the terms into additive and subtraction terms, however I'm not sure how to do this ...
6
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1answer
164 views

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Prove that for a strictly increasing natural sequence $(n_k) $ satisfying $\lim_{n \to \infty} n_k^{1/2^k}=\infty$, $\sum_{k=1}^{\infty} 1/n_k$ is irrational. This is another problem "problems in ...
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0answers
101 views

about a theorem of weakly lower semicontinuous functions

I am studying the proof of the following theorem Theorem: Let $E$ a Hilbert space and suppose that $\varphi :E \rightarrow R$ is a weakly lower semicontinuous functional. Suppose that $\varphi$ is ...
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1answer
74 views

Infinite product whose entries tend to 1 rapidly

Does the following infinite product converge and what is the limit if it exists? $$\prod_{i = 1}^{\infty} \frac{2^i}{2^i+1}$$
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1answer
31 views

To prove : $\lim_{t \rightarrow \infty} g_t \ast f = 0$.

Let $g_t(x) = e^{-x^2/2t^2}/t\sqrt{2\pi}$. To prove that for $f \in C_0(\mathbb{R})$, $$ \lim_{t \rightarrow \infty} g_t \ast f = 0.$$ Is the question valid ? Any hints on how to solve it ?
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2answers
85 views

$\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
7
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2answers
217 views

Every $x \in (0,1]$ can be represented as $x = \sum_{k=1}^{\infty} 1/{n_k}$, such that $n_{k+1}/n_k\in \{2,3,4\}$

Show that every $x \in (0,1]$ can be represented as $x = \sum_{k=1}^{\infty} 1/{n_k}$, where $(n_k)$ is a sequence of positive integers such that $n_{k+1}/n_k\in \{2,3,4\}$. Please do NOT reveal the ...