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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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Show that if $\sum x_n y_n$ converges for all $y=\{y_n: n\}$ in $\ell_2$ , then $x=\{x_n :n\}$ is in $\ell_2$.

Show that if $\sum x_n y_n$ converges for all $y=\{y_n: n\}$ in $\ell_2$ , then $x=\{x_n :n\}$ is in $\ell_2$. I am not able to go anywhere with this problem. Please help
Oliver
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How to Plot $\sqrt{\frac{a^2+(b-1)^2}{a^2+(b+1)^2}}=2$

How to plot this complex division? $$ \sqrt{\frac{a^2+(b-1)^2}{a^2+(b+1)^2}}=2 $$
user115961
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Union of a set that is not compact with 0

I know that $$A=\{ \frac{1}{n}: n \,\epsilon \, \mathbb N \}\, \subseteq\, \mathbb R $$ is not compact However, I am confused why $$A\, \cup\, \{0\}$$ is compact. My attempt at understanding: Let $B=A \cup \{0\}$ and suppose $U_n$ is an open cover…
user115862
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Pointwise Convergence with Boundedness implies $L^2$ Convergence

If $f$ and $f_k$ are integrable functions to $\mathbb{R}$ on an closed interval and $\{f_k\}$ converges pointwise to $f$ with $\sup_{k\in\mathbb{N}}\Vert f_k\Vert_\infty<\infty$, I think $$\lim_{k\rightarrow\infty}\Vert…
Analysis
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How prove or disprove $f$ Non-differentiable points countably infinite

Today,my frend ask this follow question:and I consider sometime,and I can't solve it. I hope see someone can help me Question: let $f$ is continuous strictly increasing function, prove or disprove :the $f$ Non-differentiable points countably…
math110
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Application of Contraction Mapping Theorem

How can I use the contraction mapping theorem to show there is a unique solution to $x=\cos x$ and get a reasonable estimation? Im struggling to use the theorem in this particular case so any help will be great!
Raul
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Differentiability involving Det function

I've been struggling on this for a while, and so any help will be great. I've just dont seem to handle matrix manipulation well and im not sure if my first part to the question is correct (rigour wise), onwards I'm not really advancing.
WhizKid
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Baby rudin chapter 6 exercise 14 ---Isn't it a typo?

$$ f(x) \ = \ \int_{x}^{x+1} \sin(e^t) \, dt. $$ Show that $$ e^x | f(x) | \ < \ 2 $$ and that $$ e^x f(x) = \cos(e^x) - e^{-1} \cos(e^{x+1}) + r(x), $$ where $|r(x)|< C e^{-x}$ for some constant $C$. Using Integration by parts, I showed…
user108219
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Proving Riemann Integrability- Uniform Convergence

Ok so I have come across a proof to show if we have a sequence of functions $f_n$ converging uniformly to $f$ say in the reals, such that if $f_n$ is riemann integrable then so is $f$. In the proof I've come aross there are two "obvious"…
Raul
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How to prove this property of limit of sequences

I read a property about limit of sequences as: suppose $ a_{n} \leq b_{n}, \lim _{n \rightarrow \infty } a_{n}=a, \lim _{n \rightarrow \infty } b_{n}=b $, then $a \leq b$. I know a proof by contradiction and I am wondering how to prove this…
Brain Zhang
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condition of changeability of function

Let $(f_n)$ be a sequence in $\Bbb R \to \Bbb R$ that converges to a continuous function $f(x)$. Is it true that $\lim_{x \to a} f(x) = \lim_{n \to \infty} f_n (a)$?
user109363
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How do I prove that $d(x,y)=(|x-y|)^{\frac{1}{2}}$ is a metric?

Let $X$ be a metric space with metric defined by $$d(x,y)=\sqrt{|x-y|}$$ where $x, y\in X$. How do I prove the triangle inequality for the metric $d(x,y)=\sqrt{|x-y|}$?
La Rias
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Good Textbook on Topology

I have one year calculus and one year linear algebra background. In addition, I have had one semester study in metric space analysis. Can anyone suggest some good textbooks on topology, please? A reader-friendly text with plenty of examples would be…
LaTeXFan
  • 3,548
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Intro analysis - contraction mappings

A function $ f: \mathbb{R} \rightarrow \mathbb{R} $ is called a contraction mapping if there exists a positive constant K < 1 such that $ |f(x) - f(y)| \leq K |x-y| $ d) Suppose $f:\mathbb{R} \rightarrow \mathbb{R} $ is a contraction mpaping and…
DH.
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Characterizing Logarithm functions

Assume that $f:(0,\infty)\to\mathbb{R}$ is a differentiable monotone function satisfying $$f^{-1}(f'(x))=e^{1/x},\ \forall\ x\in (0,\infty)\tag{1}$$ If $f(x)=\log_a{x}$ for $a>0$ and $a\neq 1$ then, $f$ is a solution of $(1)$. My question is: Is…
Tomás
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