(See title.) I think that I know how to prove $\implies$: If $(x_n)$ is a bounded sequence in $\ell^p$, then there is a $c\in\mathbb{R}$ such that $\|x_n\|\leq c$ for all $n$. So $$\|T_a x_n\|=\|a_n x_n\|\leq c\|a_n\|\to0,$$ from which it follows that any subsequence of $(T_a x_n)$ in $\ell^p$ converges (to $0$). (Is this correct?) However, for the converse, I have no idea. Any suggestions are greatly appreciated!
Let $E=l^p=\left\{x=\left(x_n\right): x_n\in\mathbb{R}, \sum_{1}^{\infty}\left|x_n\right|^p&lt;\infty\right\}$ with $1\le p&lt;\infty$, $\left\|x\right\|_E=\left(\sum_{1}^{\infty}\left|x_n\right|^p\right)^{1/p}.$ Let $\left(\lambda_n\right)$ is a bounded sequence in $\mathbb{R}$ and consider the operator $T$ defined by $$T(x)=\left(\lambda_1x_x,\ldots,\lambda_nx_n,\ldots\right)$$ where $x\in l^p$. Prove that $T$ is a compact operator from $E$ to $E$ iff $\lambda_n\rightarrow 0$.  Can anyone give me some hints to solve this problem? Thank you!
View approximate number of users online Is there any way to view the number of users online at any given time (or even an approximation)?  This could be really useful to provide an indication of whether your question is likely to get answered any time soon.  It could also provide a bit of an indication for why your question is getting very low views - if there are only a few users online, then that could be the reason for little interest in a question, whereas if there are lots of users online and the question still doesn't garner many views, then maybe it needs to be rephrased or something.  I know that this can't be used definitively, but it could be used as a rough guide, and could also just be an item of interest.  Perhaps it could show the number of users that have been seen in the past 5 / 10 minutes? (Shouldn't be too hard, since each user page already shows the last time that they were seen). Origin of the dot and cross product? Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner workings of them... I could get the cross product equation as a determinant of a carefully-constructed matrix,   but what I want to ask is... How did the dot and cross product come to be? When were they "invented"? Some detailed proofs? Did someone say: "Hey, wouldn't it be nice if we could construct a way to calculate a vector that is perpendicular to two given operands?"  Basically, how/why do they work?  I would appreciate explanations, links to other explanations, other web resources... I've been searching the Internet lately for explanations, but most of them are on how to use it and nothing that really gives substance to it. How do I install Ubuntu Server (step-by-step)? Intuition of pdf of a continuous random variable What is the intuition behind the probability density function of a continuous random variable? Integrating it within two points provides the probability that is associated between two points, but if you plug a single value into $f(X=x)$, it outputs a value. What is the intuitive understanding of this value for a given x?   For example, in the above image, the value of the red curve at x=0 is 0.4. How can 0.4 be interpreted? Looking for a short story title about a "weird house" Can I use Siri with these apps? Could YouTube, Todoist, myEmail (fill in name app your looking for) be controlled/used with Siri?  Could a third party iOS app be controlled by Siri? Finding files by only knowing extension? Dynamically Create a generic type for template I'm programming WCF using the ChannelFactory which expects a type in order to call the CreateChannel method. For example:   IProxy proxy = ChannelFactory&lt;IProxy&gt;.CreateChannel(...);   In my case I'm doing routing so I don't know what type my channel factory will be using. I can parse a message header to determine the type but I hit a brick wall there because even if I have an instance of Type I can't pass that where ChannelFactory expects a generic type.   Another way of restating this problem in very simple terms would be that I'm attempting to do something like this:  string listtype = Console.ReadLine(); // say "System.Int32" Type t = Type.GetType( listtype); List&lt;t&gt; myIntegers = new List&lt;&gt;(); // does not compile, expects a "type" List&lt;typeof(t)&gt; myIntegers = new List&lt;typeof(t)&gt;(); // interesting - type must resolve at compile time?   Is there an approach to this I can leverage within C#? Replace background while using HDRI for lighting? How can I replace the background of my scene, but keep the reflections and lighting generated from my HDRI? Get real client IP in a Servlet What's a command line way to find large files/directories to remove and free up space? Looking for a series of commands that will show me the largest files on a drive. Suppose $x&gt;0$ and we have function $f(x)=\frac{\sin x}{x}$ how can we show $$\forall n \in \mathbb{N} :|f^{(n)}(x)|\leq \frac{1}{n+1}$$ I need a hint to show this property .Thanks in advance . I tried for $n=1 ,2$  by finding maximum of $|f'| ,|f''|$ but I get stuck to show for $n$ Dropping the '-ing' in noun adjuncts I always remember many verbs ending in -ing.  Swimming club/cap and shaving foam for example. I now see increased use of swim club and shave foam.  Why has this happened, is it correct use of English ? How to recognise/thank great cabin crew? Those of us who fly a lot will know that there is a bit of a variety in cabin crew. Some (sadly more common on certain airlines) do the very bare minimum to ensure a safe flight, but nothing more. Many (especially on airlines noted for good customer service) provide a good service.  Every so often though, you come across a flight crew who go the extra mile. These are the ones always friendly, always helpful, always thinking ahead and solving the problem before it has happened. One moment they're serving you your drink, the next reassuring a little old lady about the flight, the next magicing a colouring book to turn an impending toddler crisis into a happy small child, then answering queries on arrivals and immigration for a first time travaller, then back to offer you a top up of what you're drinking (which they've remembered exactly). Not very often found, but very much appreciated when you happen across them on your travels!  In the case when you do luck out and find yourself on a flight with a great cabin crew (who go well beyond what you normally experience as "good" on that airline/kind of airline), is there a standard/common way to recognise them at the end of the flight? Not just to thank them then and there, but also to let the airline know that this person/group are what the rest of their staff should be aspiring to. Is there any standard way, or does it vary entirely by carrier? I am lost in normalizing, could anyone guide me please.  I have a minimum and maximum values, say -23.89 and 7.54990767, respectively.  If I get a value of 5.6878 how can I scale this value on a scale of 0 to 1. Most RPGs teach you that casual violence is the best solution to all your in-game problems. This is so well established a part of the vast majority of RPGs that there are entire satire RPGs like Greg Costikyan's  and John Tynes'  dedicated to showcasing the issue. Even now that the '80's anti-RPG hysteria is past, you see more sober critiques of endemic violence in RPGs in places like . In most RPGs, PCs become inured to murder and other antisocial activities very quickly, and enter the depths of depravity that wouldn't be appropriate in the worst parts of the world. Armed robbery, mass murder, and genocide become routine parts of an adventurer's day, something only stick-in-the-mud characters with the most extremely stated ethics object to. , even though it is not properly applied to just any conflict, is a PC's friend and they generally escalate any conflict in that direction. The sophistication of the gamer mindset towards introspection on this issue can be demonstrated in that the most meaningful question usually debated about in-game violence is some variant on &quot;but should we kill the noncombatant children&quot; or &quot;can we just murder people out of hand as long as they're from a typically evil race?&quot; In the real world, we generally a priori regard anyone having to have this discussion as a monster already. (P.S. The &quot;wipe out a given evil race&quot; thing isn't the point of this question; it is only mentioned to indicate that it's only &quot;that far&quot; that usually causes ethical handwringing from players, when that is a really quite extreme case and we should be uncomfortable with casual violence a lot sooner. ) What I'd like to have is a more realistic in-game treatment of conflict.  People getting wounded and giving up, taking and ransoming of prisoners, not always escalating a fistfight to weapons, not always escalating weapon combat to killing, etc.  Heck, as I write this, I'm watching an episode of Adventure Time with my daughter and the protagonists snuck in and rescued a princess kidnapped by the Ice King and then exit right next to his sleeping form. I thought, &quot;If this were D&amp;D they'd all be carefully coordinating a coup de grace to kill him in his sleep on the grounds that he inconvenienced them.&quot;  The problem isn't limited to D&amp;D of course, sci-fi PCs are happy to neutron bomb planets for convenience too, for example. I'm not part of the &quot;D&amp;D is Satanic&quot; crowd obviously, but I frankly do have compunctions about continually playing in games where the taught behavior is uncomfortably equivalent to the worst examples of human behavior we see on the nightly news. How can I give my PCs a newfound respect for human life? How to use exclamation marks in Minecraft commands? I know that you can use exclamation marks in Minecraft commands to check for an entity that does not have a specific tag. For example, you can use /kill @e[type=!Player] to kill all entities except for the player. I have, however, encountered two problems with using exclamation marks that I can't find a solution for.  For my first problem, I've set up a series of command blocks to summon shootable fireballs above the player’s head when a certain item is being held;  /scoreboard players add @a Fireball 1 {SelectedItem:{tag:{display:{Name:"Summon Fireball"}}}} /execute @a[score_Fireball_min=1,score_Fireball=1] ~ ~ ~ /summon Fireball ~ ~2.5 ~ {direction:[0.0,0.0,0.0]} /scoreboard players set @a Fireball 0   I used a comparator and a redstone torch so that the last command would activate when the first command would deactivate, that is, when the player is not holding the item.  The problem is, that when I move far away from the setup, the redstone stops working, leaving me unable to summon any new fireballs. Instead, I'm trying to use a command block to check when anything except for item is being held;  /scoreboard players set @a Fireball 0 {SelectedItem:{tag:{display:{Name:!"Summon Fireball"}}}}   I’ve either put it at the wrong place, or you simply can’t use exclamation marks in NBT tags. Could someone tell me where I should put the exclamation mark, or in the latter case, how I can check for anything except for a specific item?  Further, I’ve tried using exclamation marks to set the score of all entities, including the ones with a blank score, but except for the ones with a score of 2 or higher, to 0;  /scoreboard players set @e[score_arrows_min!=2] Arrows 0 /scoreboard players set @a Arrows 1 {SelectedItem:{tag:{display:{Name:"Rain of Arrows"}}}} /scoreboard players set @e[type=Arrow] Arrows 2 {inGround:1b}   The code does change the scores of entities without scores to 0, but it doesn’t exclude the arrows with a score of 2. Could anyone tell me what I’m doing wrong here? What is a word to describe inefficient communication? Example 1 - When reviewing résumés I don't want someone to use 1.5 pages to describe one job.   Example 2 - When a person uses excessive description to answer a simple question. Volume of the torus with an integral Complex differential equation How do I solve $$x'=e^{it}\overline{x}?$$  This is a complex differential equation, but I don't see how to solve it.  Edit: the original ODE is given by $$(x', y')=\begin{pmatrix}\cos t&amp; \sin t\\ \sin t&amp;-\cos t\end{pmatrix}(x,y)$$ I want to show that solutions of this ODE do not remain bounded for all $t$, and the idea was to solve the complex equation.