Starred posts

Lennis Mariana

May 5 19:53

Hi, i wanna show that quotient space X/Y is complete when X
is Banach space, and Y is a closed subspace of X, and i see that we can use the open map theorem, but i don'tn understand how to use this theorem for show the completeness of a space, can you help me?

Lennis Mariana

Apr 23 00:32

Hi, i have troubles proving that $\mathbb{CP}^1 \rightarrow S^2; \ [z:w]\mapsto \frac{1}{\vert w \vert^2+\vert z\vert^2}(2\operatorname{Re}(w\bar{z}),2\operatorname{Im}(w\bar{z}),\vert w\vert^2-\vert z \vert^2)$ is a biyective function, i think is easy show that is 1-1 but onto i don't know how, can you help me?

Martin Sleziak

Feb 20 04:03

Cleo from Math StackExchange's Identity has been Revealed??

►

2

Atri De

Apr 6 12:56

What is the best introductory real analysis book of all time?

Martin Sleziak

Apr 5 21:14

@LennisMariana There is this post on the main site: Prove that $C^1([a,b])$ with the $C^1$- norm is a Banach Space.. And the other posts linked there seem to be related.

Lennis Mariana

Apr 5 20:44

Hi, i have little trouble showing that (Cˆ1([0,1]),||f||) with ||f||=max_{x\in[0,1]}{|f|,|f'|} is a banach space because i don't know how to manage this norms in the cauchy sequences to make that this converges, Do you have any ideas to help me with that?

Martin Sleziak

Feb 21 11:31

r/math: Cleo (Math StackExchange legend) mystery finally solved

Tim Raczkowski

Jan 18, 2024 23:54

Is anyone available to check a measure theory proof?

Tapi

Nov 6, 2022 15:08

is there a way to do this without using L'hopitals rule?

2

Koro

Jun 10, 2023 13:09

This was to prepare stage for creating an onto continuous map $I\to I^2$.

Martin Sleziak

Jun 10, 2023 09:27

@Koro I suppose you're using $a_n\in\{0,2\}$ here.

Koro

May 28, 2023 06:39

How to prove continuity of $f: C\to C: \sum_{n=1}^\infty \frac{a_n}{3^n}\mapsto \sum_{n=1}^\infty \frac{a_{2n}}{3^n}$, where C is Cantor set?
Take $p\in C$. Take any $\epsilon>0$. We want a $\delta>0$, $|x-p|<\delta\implies |f(x)-f(p)|<\epsilon$. Suppose that $x=\sum_{n=1}^\infty \frac{a_n}{3^n}, p=\sum_{n=1}^\infty \frac{b_n}{3^n}$. $|f(x)-f(p)|\le \sum_{n=1}^\infty \frac{|a_{2n}- b_{2n}|}{3^n}\le \epsilon$.
How to get $\delta$ from here?

Martin Sleziak

Jul 30, 2021 08:33

Just a reminder - to read MathJax/LaTeX in chat you can use the bookmarklet mentioned in this post on meta or go directly to robjohn's website: math.ucla.edu/~robjohn/math/mathjax.html

2

beerzil charlemagne

Dec 8, 2019 08:58

What are some good sources to begin multivariate calculus (or analysis) ? I have completed real analysis in one variable and calculus in one variable . I am familiar with groups and rings from Herstein's book .Are there any prerequisites I should be covering before multivariate calculus ?

3

Tapi

Nov 6, 2022 15:07

Evan Merrill

Jan 19, 2018 04:14

This is the most inactive chat group ever.

4

Simply Beautiful Art

Sep 9, 2017 20:48

And be ready to tell me what a limit is by next week?

4

Martin Sleziak

May 20, 2022 04:40

I see that there was a question on the main site about the same integral posted today: Showing $\int _{0} ^{\pi/4} \frac{\cos^{2022}(x)}{\sin^{2022}(x) + \cos^{2022}(x) } dx \approx \frac{\pi}{4}$.

Martin Sleziak

Feb 14, 2022 20:06

Getting advice on your calculus homework from somebody with Fields medal - that's quite a cool story to tell.

anonymous

Dec 28, 2021 14:06

.

Simple

Dec 15, 2021 21:55

Prove that for any $f:\mathbb{R}\to\mathbb{R}$ the set of all points where $f$ is anti-continuous is nowhere dense.

Donald Splutterwit

Oct 7, 2017 22:18

Let's not do calculus ... let talk about our favourite food ... I had roast chicken for my dinner ;-)

3

Simply Beautiful Art

Sep 9, 2017 20:48

And the first problem from here: tutorial.math.lamar.edu/Problems/CalcI/LimitsIntro.aspx

3

Simply Beautiful Art

Sep 9, 2017 20:47

Do the first problem from each part: tutorial.math.lamar.edu/ProblemsNS/CalcI/LimitsIntro.aspx

3

Martin Sleziak

Dec 10, 2018 11:35

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

2

Verónica

Apr 2, 2021 17:13

$\dfrac{\partial F}{\partial n^m}=\left(\dfrac{\partial n^{m+1}}{\partial n^m}\right)^T\dfrac{\partial F}{\partial n^{m+1}}$ where $n$ is a vector ($\mathbf n$), $\left(\dfrac{\partial n^{m+1}}{\partial n^m}\right)$ is of size $S^{m+1}\times S^m$, and $\dfrac{\partial F}{\partial n^{m+1}}$ is of size $S^m\times1$. To simplify, assume $S^{m+1}=S^m=2.$
Thus

$\dfrac{\partial n^{m+1}}{\partial n^{m}}=\begin{pmatrix} \dfrac{\partial n_1^{m+1}}{\partial n_1^{m}}&\dfrac{\partial n_1^{m+1}}{\partial n_2^{m}}\\ \dfrac{\partial n_2^{m+1}}{\partial n_1^{m}} & \dfrac{\partial n_2^{m+1}}{\partial n_2^{m}}

Donald Splutterwit

Sep 9, 2017 20:38

No no ... welcome random person ! lol :P

2

Simple

Aug 9, 2020 23:46

Let $\alpha=\{a_k\}$ be a monotone increasing sequence with $a_1\geq0$ and $a_k\to\infty$, and consider the ellipse $$E(\alpha) =\{v\in\ell_2\;:\;\sum_{k=1}^{\infty}=a_kv^2_k\leq1\}$$. Prove that $E(\alpha)$ is a compact subset of $\ell_2$.

Simple

Jul 9, 2020 00:59

oh

Calvin Khor

Jun 27, 2020 23:30

@Simple by $1_p$, do you mean $\ell_p$? If so, then a big hint is that sequences that are eventually zero are in every $\ell_p$

Martin Sleziak

May 6, 2020 04:14

@BAYMAX For example, if $f(x)=x$ then it is (strictly) increasing and $f(x^+)=f(x^-)$.

Martin Sleziak

Apr 24, 2020 14:13

I'll just add a reminder that info on MathJax in chat can be found in this post on meta or the bookmarklet can be obtained directly from robjohn's website.

Simple

Apr 24, 2020 06:21

Prove that if a complex function $f$ has a primitive on open sets $U_1$ and $U_2$ in $\mathbb{C}$ and the set $U_1\cap\,U_2$ is connected, then $f$ has a primitive on $U_1\cup\,U_2$. Show that the assumption $U_1\cap\,U_2$ is connected cannot be omitted.

Simple

Apr 15, 2020 03:23

\begin{align*}
\lim_{n\to\infty}\left(1+\frac{z}{n}\right)^n&=\lim_{n\to\infty}\sum_{k=0}^{n}\binom{n}{k}\left(\frac{z}{n}\right)^k\\
&=\lim_{n\to\infty}\sum_{k=0}^{n}\left(\frac{n!}{k!(n-k)!}\right)\left(\frac{z}{n}\right)^k\\
&=\lim_{n\to\infty}\sum_{k=0}^{n}\left(\frac{n\cdot(n-1)\cdots(n-k+1)}{k!}\right)\left(\frac{z}{n}\right)^k\\
&=\lim_{n\to\infty}\sum_{k=0}^{n}\left(\frac{n\cdot(n-1)\cdots(n-k+1)}{n^k}\right)\left(\frac{z^k}{k!}\right)\\
&=\lim_{n\to\infty}\sum_{k=0}^{n}\left(\frac{z^k}{k!}\right)\prod_{j=0}^{k-1}\left(1-\frac{j}{n}\right)

Bob Oakley

Apr 9, 2020 07:42

I try to simplify the following singular integral:

>$$\int_{R^2}\hat{p}(\xi)\hat{q}(\xi)|\xi|^{-2}d\xi= A\int_{R^2}\int_{R^2}p(x) q(y) (-\log|x-y|)dxdy$$
where $p, q\in C_c^{\infty}$ and $\int q=0$.

My way is LHS=
$$\lim_{\epsilon\to 0}\int_{R^2\setminus B(\epsilon)}\hat{p}(\xi)\hat{q}(\xi)|\xi|^{-2}d\xi$$
But why the RHS appears $(-\log|x-y|)$ which is the log potential of Poisson equation.

Simple

Apr 8, 2020 03:53

$f(x+iy)=\sqrt{|xy|}$ is not holomorphic at the origin but satisfies the Cauchy-Riemann equations.

Simple

Apr 8, 2020 03:52

In complex analysis, are Cauchy-Riemann equation and holomorphic related? Not sure if this question makes sense.

Martin Sleziak

Mar 19, 2020 06:38

I see, in this book $|S|$ is used to denote outer measure of $S$.

Tunk-Fey

May 12, 2014 10:21

Does anyone know how to evaluate
$$
\int\frac{\sin x}{1+\cos x+e^x}dx\ ?
$$

3

Simple

Feb 6, 2020 05:35

measure.axler.net

Simple

Jan 31, 2020 05:55

Suppose $(X,\mathcal{S},\mu)$ is a measure space such that $\mu(X)$ is finite. Show that if $\mathcal{A}$ is a set of disjoint sets in $\mathcal{S}$ such that $\mu(A)>0$ for every $A\in\mathcal{A}$, then $\mathcal{A}$ is a countable set.

Simple

Jan 23, 2020 07:55

Suppose $X$ is a set and $E_1,E_2,\dots,$ is a disjoint sequence of subsets of $X$ such that $\bigcup_{k=1}^{\infty}E_k=X$. Let $\mathcal{S}=\{\bigcup_{k\in\,K}E_k\,:\;K\subset\mathbb{Z}^{+}\}$.\\
(a) Show that $\mathcal{S}$ is a $\sigma$-algebra on $X$.\\
(b)Prove that a function from $X$ to $\mathbb{R}$ is $\mathcal{S}$-measurable if and only if the function is constant on $E_k$ for every $k\subset\mathbb{Z}^{+}$.

Martin Sleziak

Jan 15, 2020 10:33

I'll just add a reminder that info on MathJax in chat can be found in this post on meta or the bookmarklet can be obtained directly from robjohn's website.

Simple

Jan 12, 2020 23:40

Show that the normed vector space $(C([a,b]),\lVert\cdot\rVert_1)$ is not a complete metric space.

Simple

Jan 7, 2020 04:29

Define $f:[0,1]\to\mathbb{R}$ as $f(a)=\begin{cases}0&\;\;\text{if $a$ is irrational}\\\frac{1}{n}&\;\;\text{if $a$ is rational and n is the smallest positive integer
such that $a = m/n$
for some integer $m$.}\end{cases}$ Show $f$ is Riemann integralble

Martin Sleziak

Nov 8, 2019 07:16

All of these functions have sin/cos of $\frac1x$. Are there any functions that can be "drawn" by pen, or otherwise simpler? — SOFe 1 hour ago

Martin Sleziak

Oct 18, 2019 13:40

I guess that checking similar posts on main (see the links above) might also be useful.

Martin Sleziak

Oct 17, 2019 10:36

BTW this room just now crossed 10000 messages - assuming the numbers in the room-info are correct. (After this message it says 10006.)

schn

Oct 9, 2019 14:41

On a different note, is the solution to exercise 14 correct? When expanding $\ln{(1+y)}$, the 4th degree term is left out, but in order to find the coeffiecent $a_4$, the 4th degree term in the expansion of $\ln{(1+y)}$ contributes with $\frac{a_1^4}{4}$ to the last equation when solving for $a_4$.

Martin Sleziak

Aug 29, 2019 14:48

For instructions how to render MathJax(TeX) in chat see this post on meta or go directly to robjohn's website.

Calculus and analysis

Calculus and analysis