Let $d_1(x,y)$ and $d_2(x,y)$ be any two metrics on $\mathbb{R}^n$. Can it be shown that,  $$c\cdot d_2(x,y) \le d_1(x,y) \le C\cdot d_2(x,y)$$  for all $x,y \in \mathbb{R}^n$ for some fixed positive constants $c,C$? If not, under what conditions could such a relation hold (for a compact set it seems straightforward)? If yes, does this result hold for two arbitrary topological metric spaces as well?  Thanks!
The following is a well-known result in functional analysis:     If the vector space $X$ is finite dimensional, all norms are equivalent.     Here is the standard proof in one textbook. First, pick a norm for $X$, say  $$\|x\|_1=\sum_{i=1}^n|\alpha_i|$$ where $x=\sum_{i=1}^n\alpha_ix_i$, and $(x_i)_{i=1}^n$ is a basis for $X$. Then show that every norm for $X$ is equivalent to $\|\cdot\|_1$, i.e.,  $$c\|x\|\leq\|x\|_1\leq C\|x\|.$$ For the first inequality, one can easily get $c$ by triangle inequality for the norm. For the second inequality, instead of constructing $C$, the  is applied to construct a contradiction.   The strategies for proving these two inequalities are so different. Here is my question,      Can one prove this theorem without Bolzano-Weierstrass theorem?   UPDATE:     Is the converse of the theorem true? In other words, if all norms for a vector space $X$ are equivalent, then can one conclude that $X$ is of finite dimension?
Visiting Greece with back-to-back Schengen visas (One from Germany, other from Italy) The 1st law of thermodynamics If a gas expands adiabatically will the work done be positive or negative?  I think it will be positive as $\delta W=p dV$ and we have a positive sign due to the work done by the gas is positive as it pushes to expand. But this site has the negative of the answer I obtain.   (See adiabatic processes section) Eclipse doesn't start on Ubuntu 18.04 I installed eclipse via Ubuntu Software. When I start it, it take some seconds and then comes an error window showing the path of the error.  The following is the .log file:  !SESSION Wed May 02 12:53:28 BRT 2018 ------------------------------------------ !ENTRY org.eclipse.equinox.launcher 4 0 2018-05-02 12:53:28.587 !MESSAGE Exception launching the Eclipse Platform: !STACK java.lang.ClassNotFoundException: org.eclipse.core.runtime.adaptor.EclipseStarter     at java.base/java.net.URLClassLoader.findClass(URLClassLoader.java:466)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:566)     at java.base/java.lang.ClassLoader.loadClass(ClassLoader.java:499)     at org.eclipse.equinox.launcher.Main.invokeFramework(Main.java:626)     at org.eclipse.equinox.launcher.Main.basicRun(Main.java:584)     at org.eclipse.equinox.launcher.Main.run(Main.java:1438)     at org.eclipse.equinox.launcher.Main.main(Main.java:1414)   What is the problem and how can I fix it? Double decimal formatting in Java Changing geometry types from kml points Z to just points in QGIS How do I change geometry types from pointZ to point in QGIS from a kmz file? Kolmogorov-Smirnov two-sample $p$-values I am using the Kolmogorov&ndash;Smirnov two-sample test to compare distributions, and I noticed a $p$-value is frequently reported as the test statistic. How is this $p$-value determined? I know it's the probability of obtaining a result at least as large as the one obtained, but how is this $p$-value determined given this is a nonparametric test? That is, we can't assume Gaussian fluctuations in the distribution and compute the $p$-value using a $t$-test.  Thanks! How can I find maximum power transfer using Thevenin equivalent Correct MySQL username/password, but getting Access Denied error when run from script 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.) Understanding Group Permissions I'd appreciate help with a couple questions regarding standard file permissions. Assume that I'm not using access control lists.     If a file is associated with multiple groups, must all groups have the same permissions?   If the group permission includes write access, can any member of any group the file is associated with change the file permissions? Can such a person change the file owner?   Say now the file permission is rwxr----- (owner can do everything, but group members can only read). I am in the group associated with the file. I am also the file owner. Which permission am I granted? Read-only, or rwx? How does the reparameterization trick for VAEs work and why is it important? How does the  for variational autoencoders (VAE) work? Is there an intuitive and easy explanation without simplifying the underlying math? And why do we need the 'trick'? Why are .goutputstream-xxxxx files created in home folder? I find a lot of files with names of the like  .goutputstream-xxxxx   where xxxxx are some alphanumeric characters.  Here is a screenshot of the home folder    Any idea why I should keep these or can I remove and also stop further creation? Proof by induction that $ 169 \mid 3^{3n+3}-26n-27$ $ 169$ | $3^{3n+3}-26n-27$ ?  Fulfilled for $n=0$. Induction to $n+1$: An integer $x$ exists so that  $ 169x= 3^{3n+6}-26n-27-26$ $ 169x= 27*3^{3n+3}-26n-27-26$ $ 169x= 26*3^{3n+3}+3^{3n+3}-26n-27-26$  An integer $m$ exists so that   $ 169x= 26*3^{3n+3}+169m-26$ ($ 13x= 2*(3^{3n+3}-1)+13m$)  Now I'm stuck although it looks simple. Thanks for any input in advance. I am writing my thesis,  and i am in restructuring phase - sorting theorems where they should belong thematically. And it would be helpful to have all the proofs hidden so i am not laden with the additional information.  Is it possible to add something like hideproof to documentclass options to omit them when compiling the latex file. centos: using epel repository on a server machine Are there any downsides in enabling epel repository on a server machine or is it something every sysadmin does? Prove that $\int_0^\infty \frac{x\,\log x}{(1+x^2)^2} = 0$ I have been asked to prove that $$\int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx = 0$$  I tried proceeding as follows: \begin{align} \int_0^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_1^\infty \frac{x\,\log x}{(1+x^2)^2}\,dx \\ &amp; = \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx + \int_{u=1}^0 \frac{u\,\log u}{(1+u^2)^2}\,du\quad (\text{where }u=1/x) \\ &amp;= \int_0^1 \frac{x\,\log x}{(1+x^2)^2}\,dx - \int_0^1 \frac{u\,\log u}{(1+u^2)^2}\,du \\ &amp;=^? \int_0^1 0 \,dx \\ &amp;= 0 \end{align}  But I think that I should first show that the improper integral $\int_0^1 {x\,\log x}/{(1+x^2)^2}\,dx$ is convergent.  Do I need to show it? If yes, how to do so?  (I have been taught basic methods such as comparison, limit comparison and Dirichlet test.) Android: TextView automatically truncate and replace last 3 char of String How can I ensure that old documents will compile in the future? Why do tuning forks have two prongs? I believe the purpose of a tuning fork is to produce a single pure frequency of vibration.  How do two coupled vibrating prongs isolate a single frequency?  Is it possible to produce the same effect using only 1 prong?  Can a single prong not generate a pure frequency?  Does the addition of more prongs produce a "more pure" frequency?  The two prong system only supports a single standing wave mode, why is that? dist$(x,A)=0$ if and only if $x\in \overline{A}$ (closure of $A$) Let $(X,d)$ be  a metric space and non-empty $A\subset X$. The distance from $x$ to $A$ is defined as $$\mbox{dist}(x,A)=\inf\{d(x,a):a\in A\}$$  I want to show that dist$(x,A)=0$ if and only if $x$ is in the closure of $A$.     (=>): dist$(x,A)=0$ implies $\inf \{d(x,a):a\in A\}=0$. Suppose $x\not\in \overline{A}$, then $x\in X\setminus \overline{A}$, which is an open set. Thus $\exists\epsilon &gt;0$ such that punctured open ball $B(x;\epsilon)\setminus\{x\}= \{y\in X:0&lt;d(x,y)&lt;\epsilon\} \subset X\setminus\overline{A}$. But then dist$(x,A)=\inf\{d(x,a):a\in A\}\ge \inf\{d(x,y):y\in B(x,\epsilon)\setminus\{x\}\}&gt;0$. This is a contradiction! Hence, $x\in \overline{A}$.      (&lt;=): Suppose $x\in \overline{A}$. If $x\in A^\circ$ (the interior of $A$) then $\inf\{d(x,a):a\in A\}=d(x,x)=0$.   If $x\not\in A^\circ$ then $\exists $ a sequence $(a_k)_k\subset A$ such that $a_k$ converges to $x$. Then, $\forall \epsilon &gt; 0, \exists N\in\mathbb{N}$ such that $k\ge N\implies d(a_k,x)&lt;\epsilon$.      So that dist$(x,A)=\inf\{d(x,a_k)\}=\lim\limits_{k\to\infty} d(x,a_n) = 0$.   Please let me know if my proof is rigorous / correct. I'm not completely sure, think it's better to check.