Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [analysis]

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).

Mathematical analysis is the rigorous version of calculus. In fact, it investigates the theorems in calculus with enough care and deals with them more deeply, trying to generalize the ideas in calculus. You can consider a more specific tag instead: , , , , , , etc. For data analysis, use .

42884 questions
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A bounded sequence whose sequence of averages does not converge

Can we find a bounded sequence $\{a_n\}$ such that the sequence of its averages, say, sequence $\{b_n\}$, where $$b_n=\frac{1}{n}\sum_{i=1}^n a_i,$$ does not converge?
OnoL
  • 2,685
7
votes
1 answer

$\|D_{f}(x) v\|=\|v\|$ $\implies$ $f$ is an isometry

Let $f\colon \mathbb{R}^m \to \mathbb{R}^m$ be a $C^2$ map such that $\|D_f(x)v\|=\|v\|$ for all $v\in\mathbb{R}^m$, where $D_f(x)$ is the derivative of $f$ at $x$. Then I am asked to prove that $\|f(x)-f(y)\|=\|x-y\|$ for all $x,y\in\mathbb{R}^m$.…
Oddone
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7
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What is the infinity norm on a continuous function space?

As I understand it the norm $\|f\|_2$ on the set of continuous functions on $[0,1]$ is defined by $$\|f\|_2 = \sqrt{\int_0^1|f(t)|^2dt}$$ but what is the infinity norm $\|f\|_\infty$? Is it $$\int_0^1 \max|f(t)|dt$$ so in other words just…
user26069
6
votes
1 answer

How find this limit $\lim_{n\to\infty}\left(\sqrt{n}\int_{0}^{1}(e^x(1-x))^ndx\right)$

Question: Find this limit $$\lim_{n\to\infty}\left(\sqrt{n}\int_{0}^{1}(e^x(1-x))^ndx\right)$$ my idea: since…
math110
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6
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1 answer

How prove there exist $(a,b)$ such $f'^2_{x}(a,b)+f'^2_{y}(a,b)-4(a^2+b^2)=0$

Question: let $D=\{(x,y):x^2+y^2<1\}$,and $f\in C^{1}(D)$,if $$|f(x,y)|\le 1 ,((x,y)\in D)$$ show that: $\exists (a,b)\in D,$$$f'^2_{x}(a,b)+f'^2_{y}(a,b)-4(a^2+b^2)=0$$ I only solve this problem: let $D=\{(x,y):x^2+y^2\le 1\}$,and $f\in…
math110
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6
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3 answers

$f\in C[0,\infty]$ and $\lim\limits_{x\to \infty}f(x)=L<\infty$. Compute $\lim\limits_{n\to \infty} \int_{0}^{2} f(nx)dx$

I'd really love your help with this Let $f$ be a continuous function in $[0,\infty )$ and assume that $\lim\limits_{x\to \infty} f(x)=L<\infty$. I need to compute: $$\lim_{n\to \infty} \int_{0}^{2} f(nx)dx.$$ Because of the fact that $f$ is…
Jozef
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1 answer

Exact definition of convergence

Let us consider a sequence $x_n$. Now let it converge to a limit $L$. Now which one of the following is the correct definition of convergence? A sequence $x_n$ is said to be convergent to a limit $L$ if given any integer $n$ there exists a…
user16186
  • 541
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Estimating a finite sum of complex numbers (Rudin Lemma 6.3).

In Rudin's book, Real and Complex analysis, Lemma 6.3. states: If $z_1,...,z_N$ are complex numbers then, there is a subset $S$ of $\{1,...,N\}$ for which $$\left|\sum_{k\in S} z_k\right|\ge \frac{1}{\pi}\sum _{k=1}^N |z_k|.$$ In the proof he…
Tomás
  • 22,559
6
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2 answers

Showing that a real function is convex

How can I show that if $\varphi$ is a real function such that $$\varphi \left(\int_0^1 f\right)\leqslant \int_0^1 \varphi (f)$$ for any Borel-measurable real function $f$, then $\varphi$ is convex. Ps: I realize homework questions have to be…
Colin
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Is the supremum of a function squared the square of its supremum

Let $f$ be a holomorphic function on the open unit disc. Is $(\sup \vert f \vert)^2 = \sup (\vert f\vert^2)$?
Fozad
  • 254
6
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1 answer

Divergence of the sequence $\sin(n!)$

Does the sequence $\sin(n!)$ diverge(converge)? It seems the sequence diverges. I tried for a contradiction but with no success. Thanks for your cooperation.
M.R. Yegan
  • 672
6
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Rudin Theorem 1.20

Can someone please explain help which proposition or axiom the following step comes from. I will highlight it in the proof. $1.20\ (a)$ If $x \in R,\ y\in R$, and $x > 0$, then there is a positive integer $n$ such that $nx > y$. Pf Let $A = \{nx\…
zzz2991
  • 419
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2 answers

Constructing reals: Prove $i$ not real

So I need to prove, from the definition of reals as Cauchy sequences of rationals, that $i$ is not a real number. The guidance given is to assume that $a\sim b$ are equivalent Cauchy sequences of rationals such that $\displaystyle…
Addem
  • 5,656
6
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2 answers

Something that isn't continuous can be proven to be continuous (so it is continuous - definitions - but doesn't look it!)

I'm sorry to post this, either I am right and it is continuous, or because I am on $\mathbb{Q}$ not $\mathbb{R}$ that saying "if that delta works, any smaller delta will!" (which can be proven by by some * value theorem) does not work. Consider…
Alec Teal
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Prove $ \left |\sin(x) - x + \frac{x^3}{3!} \right | < \frac{4}{15}$

Prove $ \left |\sin(x) - x + \dfrac{x^3}{3!} \right | < \dfrac{4}{15}$ $\forall x \in [-2,2]$ By Maclaurin's formula and Lagrange's remainder we have $\sin(x) = x - \dfrac{x^3}{3!} + \dfrac{\sin(\xi)}{5!}x^5$ for some $0<\xi<2$ subbing this in we…
Warz
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