I'm reading Partial Differential Equations, by L. Evans, and in section $4.3$ is a definition of the Fourier transform for functions in $L^2(\mathbb{R}^n)$. He starts considering a function $u\in L^2(\mathbb{R}^n)$, and a sequence $\{u_k\}_{k=1}^\infty \subset L^1(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$, with $u_k\rightarrow u$ in $L^2(\mathbb{R}^n)$.  So my question is: why there exists such sequence $\{u_k\}_{k=1}^\infty$ for every function in $L^2(\mathbb{R}^n)$? Is $L^1(\mathbb{R}^n)\cap L^2(\mathbb{R}^n)$ dense in $L^2(\mathbb{R}^n)$? Thank you!
In Lieb and Loss's Analysis, I saw that they mentioned  $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$ dense in $ L^2(\mathbb{R}^n)$ (dense wrt the $L^2$ norm, I think). But I didn't find its proof in the book. So I wonder why it is?    Does this conclusion hold for $L^1(\Omega, \mathcal{F}, \mu) \cap L^2(\Omega, \mathcal{F}, \mu)$ for any measure space $(\Omega, \mathcal{F}, \mu)$?  Are there similar statements if replace $L^1$ with $L^p$, and  $ L^2$ with $L^q$ for $p \leq q \in (0, \infty ]$ or $\in [1, \infty]$?  Thanks and regards!
I got this question in my maths paper     Test the condition for convergence of $$\sum_{n=1}^\infty \frac{1}{n(n+1)(n+2)}$$   and find the sum if it exists.   I managed to show that the series converges but I was unable to find the sum. Any help/hint will go a long way. Thank you. Disabling middle mouse button How can I make LightDM the default display manager? I have LightDM installed and can start it with sudo start lightdm. But at boot, GDM is still the default. How can change this so LightDM is loaded automatically at boot instead of GDM? I tried writing a shell script which can do automatic login into a ssh server using password which is mentioned in the script. I have written the following code:  set timeout 30 /usr/bin/ssh -p 8484 root@172.31.72.103 expect  {    "root@172.31.72.103's password"     {       send "password\r"     } }   This code is not running properly, still it is asking for the password. Can somebody please help me in solving this How can I check if a program exists from a Bash script? I have a late 2013 27" iMac running  Yosemite, and it randomly shuts off every now and then. I tried everything Reset SMC. Diagnostics, reformat Macintosh HD but still occurs and running out of options. MySQL: "UPDATE command denied to user ''@'localhost'" revtex4-1 and colortbl doesn't fill cells The MWE below shows that the \cellcolor command doesn't properly fill a table cell in revtex4-1 where it would do so in a simple article. Can anyone help me get fills that extend to the boundaries of my cells in revtex4-1?    \documentclass{revtex4-1} \usepackage{colortbl} \begin{document} \begin{tabular}{ |c|l|l| } \hline \rowcolor{red} A    &amp; B     &amp; C     \\ \hline Hello World         &amp; other     &amp; \cellcolor{blue}stuff\\ \hline \end{tabular} \end{document} Restoring a backup to an older version of SQL Server How did this question get a 'linked' question that isn't mentioned in comments or answers? Looking at  which has been closed as a duplicate (quite right), we see three 'linked' questions in the sidebar; however, only two of those questions are given as duplicate-close-targets.  which to my mind is the actual 'exact duplicate', isn't mentioned anywhere else except in the 'linked questions' sidebar itself! The  says:  The Linked sidebar... gathers up any links to this question provided by the community via comments, answers, or questions  So how did  get into the linked questions sidebar of ? (There's no explicit link in the other direction, in case you were wondering.) Is there a setup where I can leave the water on all summer? Setting Blend Width on Mosaic Dataset using ArcMap Visiting Greece with back-to-back Schengen visas (One from Germany, other from Italy) I've recently been learning about differential equations, and my teacher has been giving some particularly difficult examples out for those of us who finish early. He gave us the following differential equation: $$x=y\frac{dy}{dx}-\left(\frac{dy}{dx}\right)^{2}$$ With the hint "Differentiate with respect to $y$, then let $\frac{dy}{dx}=p$". I've solved it after the lesson and would like someone to check whether or not I've done so correctly.    Following the hint, $$\frac{dx}{dy}=\frac{1}{p}=p+y\frac{dp}{dy}-2p\frac{dp}{dy}$$ Which rearranges to $$p-\frac{1}{p}+y\frac{dp}{dy}=2p\frac{dp}{dy}$$ Multiplying through by $\frac{dy}{dp}$, $$\left(p-\frac{1}{p}\right)\frac{dy}{dp}+y=2p$$ Rearranging into the form $\frac{dy}{dp}+f(p)y=g(p)$, and writing $p-\frac{1}{p}=\frac{p^{2}-1}{p}$ $$\frac{dy}{dp}+\frac{p}{p^{2}-1}y=\frac{2p^{2}}{p^{2}-1}$$ Our integrating factor is $$e^{\int\frac{p}{p^{2}-1}dp}=e^{\frac{1}{2}\ln(p^{2}-1)}=\sqrt{p^{2}-1}$$ So we have $$\sqrt{p^{2}-1}\frac{dy}{dp}+\frac{p}{\sqrt{p^{2}-1}}y=\frac{2p^{2}}{\sqrt{p^{2}-1}}$$ The left hand side is, by design, $\frac{d}{dp}(y\sqrt{p^{2}-1})$, and the right hand side can be integrated as follows: $$\int \frac{2p^{2}}{\sqrt{p^{2}-1}}dp=\int\frac{2(p^{2}-1)+2}{\sqrt{p^{2}-1}}dp=\int 2\sqrt{p^{2}-1}dp+\int \frac{2}{\sqrt{p^{2}-1}} dp$$ These can both be solved using the substitution $p=\cosh(u)$, and gives $$y\sqrt{p^{2}-1}=p\sqrt{p^{2}+1}-\cosh^{-1}(p)+\alpha$$ Where $\alpha$ is an arbitrary constant. Dividing, we finally have $$y=p+\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}$$  Here I was stuck for a long time, until I realised that we also have the equation we began with: $x=py-p^{2}$, which we can solve as a quadratic in $p$ to also get $$p=\frac{y\pm \sqrt{y^{2}-4x}}{2} \implies p^{2}=\frac{y^{2}+y^{2}-4x \pm 2y\sqrt{y^{2}-4x}}{4}$$  Thus, $$y=\frac{y\pm \sqrt{y^{2}-4x}}{2}+\frac{\cosh^{-1}\left(\frac{y\pm \sqrt{y^{2}-4x}}{2}\right)+\alpha}{\sqrt{\frac{y^{2}-2x \pm y\sqrt{y^{2}-4x}}{2}-1}}$$  This is as good as I can do - we have a relation between $y$ and $x$ - am I done? Is this the correct answer? Is there an easier way?     Thank you for your time.    Following the suggestion by @Valentin $$\frac{dy}{dp}=1+\frac{\frac{1}{\sqrt{p^{2}-1}}\sqrt{p^{2}-1}-p\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}=1+p\frac{1-\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}$$ Hence, $$dx=\frac{dy}{p}=dp\left(\frac{1}{p}+\frac{1-\frac{\cosh^{-1}(p)+\alpha}{\sqrt{p^{2}-1}}}{p^{2}-1}\right)$$ Importance of series resistor at output of Rectifier Why does "Size on disk" vary when "Size" does not with the same set of files? Remove vote UI elements beside own question and answer posts Currently, the system makes you think you can upvote/downvote your own question/answer but then you get notified that you can't. Why have the up/down vote buttons there then?   In general, if some operation is not allowed, then don't put UI for that up or make it disabled so it is apparently disabled - no post operation error panels please?  ps: hope this is not a dupe, but if so, just comment it and I will remove  (+a joke: a user notification panel message: I deleted this file. Is this a problem? Yes/No.) What kind of matter are black holes made of? Imagine a black hole originally formed from, for example, Rubidium atoms. On the other hand, one made from, for example, Helium atoms. Will it be there any difference between the two? Or perhaps once formed it all turns into the same type of matter?  Can the "composition" of the matter-energy in the black hole be inferred somehow?  Could one say that "black hole substance" is a distinct form or state of matter different from other forms or states of matter? With Ubuntu One shut down, is there a way to sync applications between computers? Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number. I am not sure if I am doing this right, but I have this:  There exists an integer $k$. $2k =$ positive number  $(2k)^3 - 2k$ [*And this is where I get lost. How does one prove this?]