Let $f:\mathcal l^{\infty} \to \mathbb R$ be a linear functional such that $f(x)\ge 0$ whenever $x \in \mathcal l^{\infty} $ is a sequence with non-negative terms ; then is $f$ continuous ?
This is a small exercise that I just can't seem to figure out. When I see it I'll probably go 'ahhh!', but so far I haven't made any progress.  I'd like to prove that any linear functional $\phi$ on $\ell^\infty$ that is positive, i.e. for any $(x_n) \in \ell^\infty$ such that $x_n \geq 0$ for all $n \in \mathbb{N}$ we have $\phi((x_n)) \geq 0$, is a bounded linear operator and has $\|\phi \| = \phi((1,1,...))$.  Nothing I have tried worked so far, so any hints would be appreciated, thanks!
"Reality" of length contraction in SR Is this statement true?      In a metric spase $(E,d)$, a sequence $(x_n)$ is Cauchy if and only if $  \forall k\in \mathbb{N}, \lim_{n\rightarrow+\infty}d(x_{n+k},x_n)=0$   I proved that $\Rightarrow$ is true,indeed if $(x_n)$ is Cauchy then for all $\varepsilon&gt;0,\exists n_0\in\mathbb{N}, \forall p,q\geq n, d(x_p,x_q)&lt;\varepsilon$ it still right for all $ n\geq n_0$ and $m=n+k&gt;n\geq n_0$   But I don't know if $\Leftarrow$ is true ? Eigenvectors of a circulant matrix I have the Xiaomi Mi A3 and I'm currently on both phone and PC latest stock update. I am using Windows 10 20H2 and I have tried everything – but still don't get it. I tried another Mi A3 and it got attached normally. I've tried installing all kind of drivers and different ports, nothing. When suddenly it worked and I installed the stock ROM – but because the stock ROM is kinda laggy and ugly I want to change it. But I can't because I am also on stock recovery and I can't install another recovery because my device isn't recognized in fastboot mode. My device is recognized in adb mode and in edl mode as well, only not in fastboot. Converting WKT to SHP? Touchpad stopped working 20.04 Do I have to enter the UK on my UK passport? I'm a British National (Overseas), which basically means that while I don't have the right of abode in the UK, I am permitted to visit the UK without a visa. However, my passport has expired, and while it's being renewed, the schedule is a bit tight.  At the same time, I'm an US Citizen, and since I'm only planning to visit the UK, I don't need a visa if I enter on my US passport.  I'm not familiar with UK immigration law, and given that this probably isn't too common a situation, the UK Border Agency website doesn't have an answer that I can can find, nor do they seem to have a contact form that fits my situation, so here's hoping someone knows—given that I'm a British National (Overseas), am I required to enter the UK on my UK passport, or can I just use my (valid) US passport? When should comments be flagged/deleted on per-site metas? We all know , but that guidance is, naturally, geared toward main sites, where we want to keep things clean and focused.  Meta, on the other hand, is a site's "second place", with some different rules and conventions.  For example, questions on main sites can be closed as primarily-opinion-based, but meta is all about opinions.  There's a "discussion" tag and such questions collect many opinions.  Meta is different.  Occasionally I see comment flags on metas and I'm never sure what to do with them.  Setting aside things that are blatantly offensive or spam (that's a no-brainer), is there such a thing as "too chatty" or a comment discussion that should be moved to chat on meta?  I am very conservative in deleting things on per-site metas (except on  posts, which like main should stay focused).  Are there any guidelines specific to per-site metas?  I'm specifying per-site metas and not MSE because MSE seems to have conventions that fall somewhere between "main site" and "per-site meta".  If the answer is the same for all metas then great, but I don't want to assume it'll be the same. I am constructing a bibliography of as much of the prior research in a field I am working in. I would like to use BibTeX to print the citation in the pdf, and then I would like to add the details of the paper after the citation. Only then would I like to print the next citation. For example:  [1] Jahan M. Dawlaty and Yunquing Chen. Measurement of the optical absorption of graphene in the terahertz. Applied Physics Letters, 2008. Research Group: Cornell Pump Wavelength: 500 nm. Probe Wavelenth: 7 microns. Relaxation Time: 50 fs. [2] Paul George and Dmitry Vekslar, Measurement of Ultrafast Carrier Dynamics in Graphene. Applied Physics Letters, 2009.  . . . and so on. Does anyone know how this could be done? Not a problem, but have been seeing questions recently of people not being offered upgrade from 18.04 to 20.04.  I thought these upgrades were offered back in June.  I upgraded to 20.04 from 19.10 and this was offered back in May.  Imagine that different versions paths have different schedules, but google does not help explaining(that I found).  Does someone know the schedules? How to add electrical fixtures to (custom) furniture? Having electrical outlets, USB charging ports, night lights, etc would be nice to have on some pieces of furniture, like side tables. Several receptacle manufacturers make combination fixtures with USB and power, for instance, or power plus a night light.   I've thought about nailing an electrical box to a (hidden) portion of the table and putting one of those receptacles in it, but what things do I need to consider doing this? Is it as simple as it sounds,  and then just fashioning some kind of plug out of 12/2 or 14/2 wire? Does the answer change if the box is concealed (though still accessible) vs it being completely visible? Doesn't Matlab work on ubuntu 16.04? What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases? Mounting Flashdrive - mounted to home directory I wanted to mount my flashdrive, and made the mistake of setting the mount path as /home/my_name. Now all of my documents, downloads, pictures, etc. are unavailable since my flashdrive has replaced my previous home folder.  When I go to terminal and type sudo umount /home/my_name I get the error message:   umount: /home/my_name: device is busy.         (In some cases useful info about processes that use          the device is found by lsof(8) or fuser(1))   Is there a way to fix this, where I can restore my old /home/my_name folder and then mount the flashdrive to a proper location? Drupal 7. Custom module. I have a content type 'RFI' with a handful of fields I created with Drupal UI. I want a user to: fill out a form &amp; click submit while drupal: saves fields to database, creates a new node of 'RFI' and populates the fields of the node with the content provided by the form's corresponding fields.  From what I understand:  Build a form with fields (taking from the 'DBTNG examples' module just for clarity):  function rfi_build_form($form, &amp;$form_state) { $form = array();  $form['add'] = array( '#type'  =&gt; 'fieldset', '#title' =&gt; t('Add a person entry'), ); $form['add']['name'] = array( '#type'  =&gt; 'textfield', '#title' =&gt; t('Name'), '#size'  =&gt; 15, ); $form['add']['surname'] = array( '#type'  =&gt; 'textfield', '#title' =&gt; t('Surname'), '#size'  =&gt; 15, ); $form['add']['age'] = array( '#type'  =&gt; 'textfield', '#title' =&gt; t('Age'), '#size'  =&gt; 5, '#description' =&gt; t("Values greater than 127 will cause an exception. Try it - it's a    great example why exception handling is needed with DTBNG."), ); $form['add']['submit'] = array( '#type'  =&gt; 'submit', '#value' =&gt; t('Add'), ); return $form; }   Here the form fields are saved to database with database API.  function rfi_build_submit($form, &amp;$form_state) { global $user;  // Save the submitted entry. $entry = array( 'name'    =&gt; $form_state['values']['name'], 'surname' =&gt; $form_state['values']['surname'], 'age'     =&gt; $form_state['values']['age'], 'uid'     =&gt; $user-&gt;uid, ); $return = dbtng_example_entry_insert($entry); if ($return) { drupal_set_message(t("Created entry @entry", array('@entry' =&gt; print_r($entry, TRUE)))); } }   Now here is how I know how to create a node of RFI type. How do I take the values of the fields above [name], [surname], [age] and have them related to corresponding fields in my content type?  function rfi_build_type($form, &amp;$form_state){ $node = new stdClass(); $node-&gt;type = 'rfi'; $node-&gt;name = ''; $node-&gt;surname = ''; $node-&gt;age = ''; $node-&gt;language = 'en'; $node-&gt;uid = 1; node_save($node); }   I was thinking that since the fields have been saved already, that I could use a database query to get these values, but intuition tells me I'm overlooking something.  Thank you! It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.  Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".   Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?  I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism. Awk: if pattern not found add some text at the end of file I didn't grow out of it.   or      I haven't grown out of it.    Which sentence is grammatically more correct? I'd rather say the second, but  I saw in the text the first one. How does one find which files are taking up 80% of the space on a Linux webserver? The drive is constantly filling up. You've hunted down all the spare and random files you can. grep'd for coredump files and even removed some of the un-needed backups...  What would be your next move.  The actual server in question has 10GB of website files and the OS shouldn't take any more than 10GB so how do you track down what's filling a 50GB (virtual) drive? Generators and relations for $\text{PSL}(2, \mathbb{F}_q)$ Is there a nice presentation for the group $\text{PSL}(2,\mathbb{F}_q)$, for every prime $q$?  (I don't have any particular definition in mind for "nice", other than, say, a small number of generators and relations.)