Question:  Why was this marked as a duplicate?  The referenced question asked nearly a year later than this question.  In fact I'm not even sure that they are identical at all.    I was working on a homework problem which stated:     Let $X$ be a normed linear space with two norms $\|\cdot\|_1$ and $\|\cdot\|_2$.  Assume that $\|x_n - x^1\|_1\to 0$ for some $x^1\in X$ if and only if $\|x_n - x^2\|_2\to 0$ for some $x^2\in X$.  Show that $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent.   At first I errantly misinterpreted the question as $x_n\to x$ in $\|\cdot\|_1$ if and only if $x_n\to x$ in $\|\cdot\|_2$.  UPDATE:  I found a way to go directly from the convergence condition to equivalence of norms.  So I don't need to prove that the sequences converge to the same limit in this way (since it now follows from equivalence of norms).  I won't erase anything though, as the answer from Andres below may be helpful to someone else.  Now that I realize my mistake I'm trying to fix the proof.  Since the conclusion I'm trying to prove (that the norms are equivalent) implies that the limits in either norm must agree, I know that must also be true, even though it's not given in the question.  After failing to prove this extra condition, I am getting suspicious that it might be needed as a hypothesis after all.  Can anyone confirm or refute this?  My attempt:  Taking $\alpha = \dfrac12\min\left(\|x^1 - x^2\|_1, \|x^1 - x^2\|_2\right)$,  I let $B_{1} = \{x\in X : ||x - x^1||_{1} &lt; \alpha\}$ and let $B_{2} = \{x\in X : ||x - x^{2}||_{2} &lt; \alpha\}$.  Now suppose for a contradiction that there is some $x\in B_{1}\cap B_{2}$.  Then $||x - x^{1}||_{1} &lt; \alpha$.  And $||x - x^{2}||_{2} &lt; \alpha$.  I want to use this to prove $||x^{1} - x^{2}||$ (in either norm) is less than $2\alpha$ (which would give me a contradiction).  But I keep ending up with these floating $||x - x^{2}||_{1}$ or $||x - x^{1}||_{2}$ terms that I can't do anything with:  $ \begin{eqnarray*} ||x^{1} - x^{2}||_{1} &amp;\leq&amp; ||x^{1} - x||_{1} + ||x - x^{2}||_{1}\\ &amp;&lt;&amp; \alpha + ||x - x^{2}||_{1} \end{eqnarray*} $  and   $ \begin{eqnarray*} ||x^{1} - x^{2}||_{2} &amp;\leq&amp; ||x^{1} - x||_{2} + ||x - x^{2}||_{2}\\ &amp;&lt;&amp; ||x^{1} - x||_{2} + \alpha \end{eqnarray*} $
I'm working on an exercise from functional analysis.  Let $E$ be a vector space and $\|\cdot\|_1$ and $\|\cdot\|_2$ be two complete norms on $E$. Now suppose that $E$ satisfies the following property:  $\bullet$ if $(x_n)$ is a sequence in $E$ and $x,y\in E$ such that $\|x_n-x\|_1\to 0$ and $\|x_n-y\|_2\to 0$, then $x=y$.  Now we want to show that the norms $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent.  My idea is as follows: If for any $n&gt;0$, there is an element $x_n\in E$ such that $\|x_n\|_1&gt;n\|x_n\|_2$. Then consider $(\frac{x_n}{\|x_n\|_1})_{n\geq 1}$. Clearly, $(\frac{x_n}{\|x_n\|_1})_{n\geq 1}$ converges to $0$. However, I cann't get a contradicition from this. Maybe my idea is wrong.   In fact, I even don't konw how to show that a Cauchy sequence in norm $\|\cdot\|_1$ is also a Cauchy sequence in norm $\|\cdot\|_2$.   Anyone can give me some hints or a counter example? Thank you very much.
How to activate GRASS GIS 7 plugin in QGIS 2.8? How do I get the last column text to be at the bottom of the row?  That is I want the XXX to be aligned with the line containing the arrow. I tried playing around with \multirow{3}{-3ex}{XXX}, but obviously I don't understand \multirow. Code: \documentclass{article} \usepackage{tabularx} \usepackage{booktabs} \usepackage{multirow}  \begin{document}     \begin{tabularx}{\linewidth}[t]{l X p{1.3cm}}         &amp; Header &amp; Title \\\cmidrule(lr){2-2}\cmidrule(lr){3-3}         a. &amp; Some text text text text text text text text text text text text text text              text text text text text text text text text text              Want XXX on this line $\rightarrow$%          &amp; XXX\\     \end{tabularx} \end{document} Is the sum and difference of two irrationals always irrational? If $x$ and $y$ are irrational, is $x + y$ irrational? Is $x - y$ irrational? Find and replace text within multiple files I want to know how I can find and replace a specific text in multiple files like in Notepad++ in the linked tutorial.  e.g.: Supersymmetry vs multiverse I'm a complete noobe in physics and quite honestly need help. My question is simple, based on CERN's tentative findings stating the Higgs boson at a mass of ~125 GeV: Is the physics community leaning more towards supersymmetry or multiverse? Show that $E(X)=\int_{0}^{\infty}P(X\ge x)\,dx$ for non-negative random variable $X$ Show that for a non-negative random variable $X$,   $$\mathbb E(X)=\int_0^\infty \mathbb P(X\ge x) \, dx.$$   I started with  $$\mathbb E(X)=\int_0^\infty x \, dF(x)=\int_0^\infty \int_0^x dt\,dF(x).$$ This is my try. How do I recover my accidentally lost Windows partitions after installing Ubuntu? I have been reading through my notes and I came across this example and I found it hard to understand so I need some help in explaining how the inverse of this is found.  The set $\mathbb Q(\sqrt 2)=\{a+b\sqrt2$: $a,b$ in $\mathbb Q \}$ is a field. The inverse of an element $a+b\sqrt 2$ in $ \mathbb Q(\sqrt 2$) is  $$\frac{1}{a+b\sqrt{2}}=\frac{a}{a^2-2b^2}+\frac{-b}{a^2-2b^2}\sqrt{2} $$  Can anyone explain  to how the inverse was found? Appreciate your help. I have looked through several questions and answers on EL&amp;U, and often there is an indication that American English prefers "have" while British English prefers "have got". In addition, there are several references to "have got" being more informal than "have" (e.g.: , ). But where is it considered more informal? In American English alone or both American and British English?   This point may seem obvious to those who made the above mentioned references, but the fact is that even the most fluent users of English in my country will say without a doubt that 'have got' is the preferred usage in the UK and, therefore, it must also be more formal. Why? Because in an academic environment, the more informal structures are generally frowned upon, outside some specific spoken exercises. And if the teachers insist on using "have got", it follows it isn't informal.  So, is the teaching of (British) English in our schools transmitting the wrong idea?  EDIT: I'm adding some information in answer to the comments.  I'm mainly concerned with the idea of possession:      I have got a cat vs. I have a cat and She has got a dog vs. She has a dog   As mentioned in the comments, every (Portuguese) student will be told that "have" and "have got", when it comes to the idea of possession, are absolute synonyms, the only two differences being:    "have got" is preferred by British English, "have" is preferred by American English   the negative and interrogative structures differ    I have never heard of any distinction in formality. In fact, I was rather surprised when I went through the EL&amp;U archives and read about it.  So, would "I have got a cat" be as formal/informal as "I have a cat"?   And how do British and American English look at its formality/informality? Is it the same? Only squares instead of letters and numbers are displayed in my calculator and other apps in Ubuntu 18.04 Schengen visa validity start date My Chinese in-laws intend to visit us in one month from now. Their visa is valid from Nov 23rd. Would it be possible for them to take the airplane on November 22nd in China and arrive in Schengen area on November 23rd?  Or does the visa already have to be valid upon departure?  In our particular case, they still need to transfer in Beijing first. The actual flight to Schengen zone effectively only departs on the 23rd so I guess that should be completely fine then? Where can I find medicine? I know it is possible to be stung by bees in Animal Crossing: New Leaf.   I learned that the hard way when I was trying to find my furniture tree. Now, I have an eye swollen shut and all of my towns people are making fun of me. They're all being kind enough to inform me that I need Medicine, but they seem to be enjoying themselves at my expense too much to tell me where to actually find the Medicine.  Where can i get medicine? I'd really like it if my townspeople stopped making fun of me. Why does pizza cheese seem hotter than the crust? When I eat hot pizza or a melted cheese sandwich, the cheese feels a lot hotter than the crust or bread: in particular, the cheese might scald the roof of my mouth. but the crust will not. Is this   my imagination, or because the crust cools a little faster than the cheese, so has already cooled a bit by the time I eat it, or because the cheese cools a little faster than the crust, so transfers heat to the roof of my mouth a bit more, or what? Remove password from an Excel Document I'm providing internal support and one of our users has managed to put a password on an excel file by accident, I've done the proper checks to make sure that the user should have access to the document and now want to know what the recommendation for removing a password from an Excel document.  For what its worth, the password appears after Excel opens but before you can see any data in excel. SHM with acceleration at mean position Suppose we have an equation of  motion as $$\frac{d^2x}{dt^2} = -kx + c,$$ then can it be called a SHM? Since acceleration is still proportional to displacement. But then, how will we define the mean position? From Awodey: In any category $\mathsf{C}$,  an object $0$ is initial if for any object $C$ there is a unique morphism $0 \to C$,  an object $1$ is terminal if for any object $C$ there is a unique morphism $C \to 1$.  First I was puzzled by the fact that we can do mappings from a zero set to any other set in a category $\mathsf{Sets}$. But then after I accepted it conceptually, I do not understand why there is no unique morphism to an empty set from every object in a category $\mathsf{Sets}$. Is this because we cannot map a non-empty set to an empty set (since our mapping is total) although we can map an empty set to a non-empty set?  Please, note, I am familiar with the proposition saying: Initial (terminal) objects are unique up to isomorphism. For every $g\in G$ there exists an $h\in G$ such that $g = h^3$ Let G be a finite group whose order is not divisible by $3$. Show that for every $g\in G$ there exists an $h\in G$ such that $g = h^3$. Question on the stability of the solar system One of the pertinent questions about many body systems that causes me much wonder is why the solar system is so stable for billions of years. I came across the idea of "resonance" and albeit an useful concept, it hardly explains the long stability of the solar system. Normally an N body problem with inverse square mutual interaction is an example of a chaotic system. Is there any real progress about this stability issue? GRUB does not detect Windows I have finally installed Ubuntu on my second drive. When I start my computer GRUB only offers me to boot Ubuntu, not Windows 7. What needs to be done so that I can choose between Ubuntu and Windows in GRUB?  When I press F12 for boot menu at startup and I choose Windows Boot Manager it boots into Windows 7.    I ran command sudo fdisk -l and here is log ():  WARNING: GPT (GUID Partition Table) detected on '/dev/sda'! The util fdisk doesn't support GPT. Use GNU Parted.   Disk /dev/sda: 1000.2 GB, 1000204886016 bytes 255 heads, 63 sectors/track, 121601 cylinders, total 1953525168 sectors Units = sectors of 1 * 512 = 512 bytes Sector size (logical/physical): 512 bytes / 4096 bytes I/O size (minimum/optimal): 4096 bytes / 4096 bytes Disk identifier: 0xc3ffc3ff     Device Boot      Start         End      Blocks   Id  System /dev/sda1               1  1953525167   976762583+  ee  GPT Partition 1 does not start on physical sector boundary. Pullback of a Volume Form Under a Diffeomorphism. I have an exercise here, which I have no idea how to do.     Problem: Let $ U $ and $ V $ be open sets in $ \mathbb{R}^{n} $ and $ f: U \to V $ an orientation-preserving diffeomorphism. Then show that   $$   {f^{*}}(\text{vol}_{V}) = \sqrt{\det \! \left( \left[   \left\langle {\partial_{i} f}(\bullet),{\partial_{j} f}(\bullet) \right\rangle   \right]_{i,j = 1}^{n} \right)} \cdot   \text{vol}_{U}. $$   Notation:         $ \text{vol}_{U} $ and $ \text{vol}_{V} $ denote the volume forms on $ U $ and $ V $ respectively.   $ f^{*}: {\Lambda^{n}}(T^{*} V) \to {\Lambda^{n}}(T^{*} U) $ denotes the pullback operation on differential $ n $-forms corresponding to $ f $.