Let $T\colon X\to Y$ be a linear operator with norm  $$\|T\|=\sup_{\|x\|=1}\|Tx\|.$$ Prove that  $$\|T\|=\sup_{\|x\|\leq 1}\|Tx\|.$$
How do you prove that these four definitions of the operator norm are equivalent? $$\begin{align*} \lVert A\rVert_{\mathrm{op}} &amp;= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\ &amp;=\sup\{ \lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert\leq 1\}\\ &amp;=\sup\{\lVert Av\rVert\;\colon\; v\in V\text{ with }\lVert v\rVert = 1 \}\\ &amp;=\sup\left\{ \frac{\lVert Av\rVert}{\lVert v\rVert}\;\colon\; v\in V\text{ with }v\neq 0\right\}. \end{align*}$$
Why so many files in /usr/share/app-install/desktop? For 14.04, I'm trying out Lubuntu LTS.  If I run ls -l /usr/share/app-install/desktop, I get a list of 3085 .desktop files. All except one are dated Apr 18. The exception is:  -rw-r--r-- 1 root root  6328 Mar 28  2013 applications.menu   From the dates, it's clear that all these are installed by default and not by me.   Doesn't each GUI-based program bring in its own .desktop file at the time it's installed (mostly in /usr/share/applications)? Then why are these thousands of other .desktop files needed? From their names, most of them haven't anything to do with Lubuntu. Their names are different from regular .desktop files (such as those in /usr/share/applications): many, not all, are like name:name.desktop whereas the regular ones are just name.desktop. The ones I looked at all have a line with X-AppInstall-Popcon=nnn where nnn is a string of digits. I know that Popcon "gathers statistics determining which packages are the most popular with Ubuntu users". But why is that part of these .desktop files? Who did that to the priest's dog? In John Michael McDonagh's wonderful philosophical black comedy , a good priest receives a death threat during a confessional giving him a week to sort out his affairs.   During the course of the week there are a lot of encounters between the priest and his parishioners, many acting as red herrings to distract the audience form the identity of the potential killer. Several threatening acts also take place: the church is burned and the priest's dog is killed.   But in the final encounter between the priest and the mystery antagonist (I'm not saying who to avoid spoilers) the antagonist admits burning the church but claims not to have killed the priest's dog.  Are they lying? Or is there a good reason that someone else might have killed the dog? How can I change iTunes country of region when having some credit in my account? I moved to Canada and I want to change my country of region from USA to Canada, but when I try to change it ( by click on my account in the iTunes Store and going to my Account Page) it showed me this error: "You have a store credit balance; you must spend your balance before you can change stores"    I must mention I have redeem 100$ to my account but I don't have any intention to spent it now, what is my option? How to set tick label precision RECENTLY ON THE ANDROID SITE WE GOT  WRITTEN IN ALL CAPS.  THIS WAS SURPRISING TO ME, SINCE I ASSUMED IT WOULD HAVE BEEN CAUGHT BY THE QUALITY FILTER &mdash; BESIDES BEING INDICATIVE OF OTHER POST QUALITY ISSUES, IT'S AWFUL TO LOOK AT AND REQUIRES A LARGE AMOUNT OF EDITING TO FIX (EVEN IF YOU RUN IT THROUGH TOLOWER() FIRST).  WE ALL KNOW THAT CAPS LOCK IS , BUT STILL: CAN WE BLOCK THESE QUESTIONS?    RICK  RICK  OH MAN THAT STUFF JACOB G  IS COMEDY GOLD  RIIICK     Note to editors: This post looks exactly like it is supposed to look, as an example of a question in all-caps. Please do not replace the uppercase letters with lowercase. Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $ Express $\sqrt{3}\sin\theta - \cos\theta$ as: $a\cos (\theta + \alpha) $  Can someone please explain to me how to go about doing this? Compare Debian (left) and Ubuntu (right):  $ ifconfig                                 $ ifconfig bash: ifconfig: command not found          eth0     Link encap ... $ which ifconfig                           $ which ifconfig $                                          /sbin/ifconfig   Then as superuser:  # ifconfig                                 # ifconfig eth0      Link encap ...                   eth0     Link encap ... # which ifconfig                           # which ifconfig /sbin/ifconfig                             /sbin/ifconfig   Furthermore:  # ls -l /sbin/ifconfig                     # ls -l /sbin/ifconfig -rwxr-xr-x 1 root root 68360 ...           -rwxr-xr-x 1 root root 68040 ...   It seems to me the only reason I cannot run ifconfig without superpowers on Debian is that it's not in my path. When I use /sbin/ifconfig it does work.   Is there any reason I should not add /usr/local/sbin:/usr/sbin:/sbin to my path on Debian? This is a personal computer, I am the only human user.    Versions used (uname -a): Ubuntu:   Linux ubuntu 3.13.0-51-generic #84-Ubuntu SMP Wed Apr 15 12:08:34 UTC 2015 x86_64 x86_64 x86_64 GNU/Linux   Debian:  Linux debian 3.16.0-4-amd64 #1 SMP Debian 3.16.7-ckt11-1+deb8u3 (2015-08-04) x86_64 GNU/Linux Evaluation of Indefinite Integral resulting in Hypergeometric Function I am attempting to derive the result:  $$ \int \left(1+x^n\right)^{-1/m}dx= x\,_2F_1\left(\frac 1m,\frac 1n;1+\frac 1n;-x^n\right)$$  First, I start off with the binomial expansion of the integrand to get:  $$\int\sum_{k=0}^{\infty}\frac{(1/m)_k}{k!}\left(-x^n\right)^kdx  $$  Then, I pull out a $-1$ and interchange summation and integration:  $$\sum_k\int (-1)^k\frac{(1/m)_k}{k!}x^{nk}dx  $$  $$ \sum_k (-1)^k\frac{(1/m)_k}{k!}\frac{x^{nk+1}}{nk+1} $$  $$ x\sum_k \frac{(1/m)_k}{k!}\frac{(-x^n)^k}{nk+1}  $$  So, I'm just right there, but not sure how to express this series in the general form of a hypergeometric series with the two additional Pochhammer symbols.  I even typed this last series into Mathematica, and it returned the Hyper. Function. What am I missing, or how do I transform this last series into the desired form? I hope somebody has already figured this one out. I just installed Geoserver 2.9 on a vanilla Ubuntu 16.04 distro. The Geoserver 2.8 method of enabling CORS with the shanbe.hezoun class does no longer work with Jetty 9.2.13.  There are mentions that CORS support is already packaged with Jetty 9.2.13 in the jetty-servlets.jar.  The Jetty lib which is compiled with Geoserver contains a jetty-servlet-9.2.13.v20150730.jar in geoserver/lib but not jetty-servlets.9.2.13.v20150730.jar. Are these supposed to be the same jar with a different name?  It should be possible to enable CORS either in geoserver/etc/webdefault.xml or in geoserver/webapps/geoserver/WEB-INF/web.xml.  My understanding is that the webdefault.xml is applied first and the web.xml thereafter.  I have tried following filter in both xml. I haven't got as far as adding a filter mapping. Adding the filter alone will cause the Geoserver/Jetty service to not start proper.  &lt;filter&gt;     &lt;filter-name&gt;cross-origin&lt;/filter-name&gt;     &lt;filter-class&gt;org.eclipse.jetty.servlets.CrossOriginFilter&lt;/filter-class&gt; &lt;/filter&gt; How do I get object names to appear in the render? In my scene I have multiple objects (30+). In the viewport I have the handy control of selecting (in the Properties Editor) Object> Display> Name and copying to selected.  This is beautiful for diagrams etc. as wherever my view is at whatever distance these labels have been placed instantly, always face me and are always the same size.  How can you get these to display in a rendered view?  Is the ONLY option to manually create a label for each object, get them to track to your camera (Probably have to track to an empty miles behind your camera rotation otherwise they offset if your camera isn't facing exactly front on to an object) and add some sort of scaling property to change depending on how far away you are? $E$ is a measurable set of finite measure and $1 \le p_{1} &lt; p_{2} &lt; \infty$. Consider the linear space $L^{p_{2}}(E)$ normed by $||.||_{p_{1}}$. Is this space Banach? Is “what’s” a correct short form of “what does”? E.g. “What’s he think?”  Usually ’s is short for “is” so I don’t know. The latex file coding is not there but the pdf file is there I wrote all my coding in the .tex file. The next day when I opened it the coding is all gone and the pdf file is opening with it and is displayed as an output. How do I bring back my coding? Is tipping mandatory at Restaurants and Bars in Germany? How do I tell Maven to use the latest version of a dependency? The indefinite article "an" is used when the following word starts with a vowel sound. For example one write "an adjective" but "a noun".   However, when an abbreviation is used, should this be based on the expanded meaning, or on the letters of the abbreviation.  Specifically, one writes "a noun phrase". However, here on ell.se, the abbreviation "NP" for 'noun phrase" is commonly used. If this were pronounced by sounding out its letters, as "En-Pee" it would call for "an". If it were read as "noun phrase"  it would call for "a". Should one write "a NP" or "an NP" here on ELL?  I incline to "a NP" and i used that in  just a few minutes ago, but I am not at all sure. Why does Li-Po battery voltage drop under load? I have very low background in electronics.  I have a Li-Po battery connected to an electric motor. The question is:   Why does battery voltage drop under load? Why is each cell of a Li-Po battery limited to 3.0 V (discharging)? Why is each cell of a Li-Po battery limited to 4.2 V (charging)? $A \cup B$ and $A \cap B$ connected $\implies A$ and $B$ are connected Let $X$ be a topological space. Let $A$ and $B$ be two subsets of $X$ such that: $A$ and $B$ are closed, $A \cup B$ is connected, $A \cap B$ is connected     Prove that $A$ and $B$ are connected.   Thoughts:  Suppose that $A$ is disconnected. Since $A$ is closed, there are two non-empty, disjoint, closed sets $E$ and $F$ in $X$ such that $A = E \cup F$. Then I attempted many things but all were not really useful.  Please help. Thanks. How to log out of a Stack Exchange site? Per the title, where is the logout button?  All the existing questions I can find refer to the previous design of the site, where hovering over your name in the top bar brought up a menu. What is the entropy of the universe today? What's the entropy of the universe today? How does one go about calculating this? I've heard the statement that black holes account for the bulk of the entropy in the universe today, but don't know why this would be true or the relationship between black holes and entropy.