In a theorem I am reading about closed subspace the author states that an infinite dimensional subspace need not be closed.  What is an example of infinite dimensional subspace that is not closed?
Are there any examples of subspaces  of $\ell^{2}$ and $\ell^{\infty}$ which are not closed?
Verb/Subject Agreement Make/Makes I have B1/B2 Visa. Can I travel for Business trip to US with this Visa? I have been given B1/B2 Visa, but first time when I had applied, it  was for vacation &amp; holiday. Now I need to go to US for business meetings. Is this Visa good enough or do I have to apply for a B1 Visa again?  Note: I originally applied for vacation purposes, but in my visa it is stamped "R b1/b2" How can I get back a system file after deleting it from my Mac? I've deleted a system file from my Mac and need to get it back.  System files can include:   hidden files in the root of / - especially unix files in /private  /System files /Library files /Applications files that ship with the core OS   System files are not any app, user files or files that can be regenerated by a known sequence such as cache files, Spotlight index files, etc…    Canonical question and answer for . I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$.  Okay, fine.  You write down the action functional, require that it be a minimum (or maximum), and arrive at the Euler-Lagrange equations.  Great.  But now you want these Euler-Lagrange equations to not just be derivable from the Principle of Least Action, but you want it to be equivalent to the Principle of Least Action.  After thinking about it for awhile, you realize that this implies that the Principle of Least Action isn't really the Principle of Least Action at all:  it's the "Principle of Stationary Action".  Maybe this is just me, but as generous as I may be, I will not grant you that it is "natural" to assume that nature tends to choose the path that is stationary point of the action functional.  Not to mention, it isn't even obvious that there is such a path, or if there is one, that it is unique.  But the problems don't stop there.  Even if you grant the "Principle of Stationary Action" as fundamentally and universally true, you realize that not all the equations of motions that you would like to have are derivable from this if you restrict yourself to a Lagrangian of the form $T-V$.  As far as I can tell, from here it's a matter of playing around until you get a Lagrangian that produces the equations of motion you want.  From my (perhaps naive point of view), there is nothing at all particularly natural (although I will admit, it is quite useful) about the formulation of classical mechanics this way.  Of course, this wouldn't be such a big deal if these classical ideas stayed with the classical physics, but these ideas are absolutely fundamental to how we think about things as modern as quantum field theory.  Could someone please convince me that there is something natural about the choice of the Lagrangian formulation of classical mechanics (I don't mean in comparison with the Hamiltonian formulation; I mean period), and in fact, that it is so natural that we would not even dare abandon these ideas? What is the symbol on the supply voltage that looks like a shaded-in capacitor? I am looking at  (dual low-voltage power amplifier). On the fourth page, there is a sample circuit; right at the V+ input there is a symbol that looks like a capacitor. However, it looks like it is shaded in. What is that? The app keeps asking for a nickname, and I have entered dozens of names, but it keeps saying that the names aren't available. How can I find an available name? How to learn commutative algebra? Test condition in if command Any idea why echo statement is being executed in both scenarios?  Outputs it worked:  #!/bin/bash  rizwan='' if [ -n $rizwan ]; then   echo "it worked" fi   Replace -n with -z, and it outputs it worked when expecting no output:  #!/bin/bash  rizwan='' if [ -z $rizwan ]; then   echo "it worked" fi I've seen videos of some folks having several armor sets (4 is the most I've seen so far) standing about on their settlement. I'm wondering if having too many would cause some sets to disappear. They're not like guns/items that you can safely store in settlement crates, the suits kind of just... stand around. (I know there's the extra step of taking out the fusion cores, so that NPC's don't use them.)  Is there a limit to the number of Power Armor sets you can keep? Arrays in Java and how they are stored in memory I have a question about deriving the coordinate representation of momentum operator from the commutation relation, $[x,p]= i$.  One derivation (ref W. Greiner's Quantum Mechanics: An Introduction, 4th edi, p442) is as following: $$ \langle x|[x,p]|y \rangle = \langle x|xp-px|y \rangle = (x-y) \langle x|p|y \rangle. $$ On the other hand,  $ \langle x|[x,p]|y \rangle = i \langle x|y \rangle = i \delta (x-y)$. Thus $$ (x-y) \langle x|p|y \rangle = i \delta(x-y). \tag{1} $$  We use $(x-y) \delta(x-y) = 0$. Take the derivative with respect to $x$; we have $\delta(x-y) + (x-y) \delta'(x-y) = 0$. Thus $$ (x-y) \delta'(x-y) = - \delta(x-y). \tag{2} $$  Comparing Eqs. (1) and (2), we identify $$ \langle x|p|y \rangle = -i \delta'(x-y). \tag{3} $$  In addition, we can add $\alpha \delta(x-y)$ on the right-hand side of Eq. (3), i.e. $$ \langle x|p|y \rangle = -i \delta'(x-y)  + \alpha \delta(x-y), $$ and $[x,p] = i$ is still satisfied. We can also add $$ \frac{\beta}{\sqrt{|x-y|}}\delta(x-y)$$ on the RHS of Eq. (3).  Here $\alpha$ and $\beta$ are two real numbers.  My question is, what is the most general expression of $\langle x|p|y \rangle$? Can we always absorb the additional term into a phase factor like Dirac's quantum mechanics book did?  Thank you very much in advance. Am I being charged taxes on my grocery purchases at Walmart? Today, I only bought groceries at Walmart and noticed that there is no tax on groceries in California (Google tells me that this is perhaps true for whole of the US?)  So far I have always paid tax at Walmart for my grocery purchases, but that also included items like crackers, shampoo etc  Did Walmart charge me then only for the items that were not groceries, or if your cart has groceries + non-grocery items, you get charged tax on all of them?  Or is this a policy that varies by store, or cashiers' knowledge/discretion, etc?  Would people recommend for me to break down a single mixed purchase into two: one only having groceries and the other with the rest, or is this being paranoid?  My grocery purchases, every two weeks, amount to around $100+.  I pay 8.25% tax, so that is enough to get me free gas if I can figure this out! css background color with floating elements 80s: the "Eighties"   90s: the "Nineties"   00s: the ???   For that matter, what is the second decade called? The "tens" just doesn't sound right. ꕺ (U+A57A Vai Syllable Kpoo) is allowed in character names as the last character I believe this should be changed so ꕺ can not be the last character of a name at minimum. This could be a bug in my opinion because it provides confusion with diamond mods.  Although other symbols are blocked from what I understand but this does technically look different.  I have changed my name to such to demonstrate the confusion at least temporarily.  They say imitation is the greatest form of flattery so maybe this should be accepted because to anyone who is familiar with SO I doubt this would cause an issue. Is there a symbol for "unitless"? I have county data in which the &quot;FIP&quot;s (actually State &quot;FIP&quot;s and County &quot;FIP&quot;s combined). I have data imported as CSV with the sames &quot;FIP&quot;s data titled 'STCounty'. However, the shapefile &quot;FIP&quot;s is listed as a Real and the CSV came in as an Integer and the Join function does not seem to work.  The drop boxes are greyed out/empty. So the bigger questions is how do you change the attribute type in a table? I tried to create a new column in the shapefile (county data) and set it to an integer and then multiply the &quot;FIP&quot;s*1, but got a column full of NULL. I am new to QGIS and have a lot of experience in MapInfo, so the join concept is not new to me, just the syntax in QGIS. Clone hard drive to another, smaller hard drive Is there a good tool/program that allows me to clone a system drive to another, smaller hard drive?  I've bought a new 64GB SSD drive and would like to move my current Windows 7 Professional 64-bit installation to that new drive. My current drive is a 200GB SATA drive with one partition only. Of course there's enough free space to shrink my current partition to less than 64GB. I had a heck of time while editing my  question. But, I guess, I discovered a new weird problem in the editor while formatting hyperlink text next to  tag.  To illustrate this problem better, I would like to give the following examples:  Has problem with one or none white-space in between:  [meta-tag:status-planned] [details page reads:][2] [meta-tag:status-planned][details page reads:][2]     Gives weird output as:   [details page reads:]   [details page reads:]   But, its okay with two white-spaces in between:  [meta-tag:status-planned]  [details page reads:][2]   Gives the correct output as:      Something is strange here. I don't understand why do I have to insert two white-spaces instead of just one? Are the two white-spaces are mandatory when using [meta-tag:some-tag] with hyperlink text adjacent to it? Or, is it one of those  thing?  But wait, it's get even further complicated if I do like:   1. No white-spaces in between:  [meta-tag:status-planned][meta-tag:status-completed][Moderator-only Tags][3]     2. One white-space in between:  [meta-tag:status-planned] [meta-tag:status-completed] [Moderator-only Tags][3]       3. Two white-spaces in between:  [meta-tag:status-planned]  [meta-tag:status-completed]  [Moderator-only Tags][3]         They all seems to work perfectly fine. Now, thats very strange! Why would the two [meta-tag:some-tag] works like a charm with or without requiring to insert any white-spaces? Can anyone explain this in a better way. I don't want to jump on the conclusion by calling this as a bug yet, so I used  tag instead of  tag. Please share your thoughts. Why isn't the gamma function defined so that $\Gamma(n) = n! $? As a physics student, I have occasionally run across the gamma function  $$\Gamma(n) \equiv \int_0^{\infty}t^{n-1}e^{-t} \textrm{d}t = (n-1)!$$  when we want to generalize the concept of a factorial.  Why not define the gamma function so that   $$\Gamma(n) = n!$$  instead?  I realize either definition is equally good, but if someone were going to ask me to choose one, I would choose the second option.  Are there some areas of mathematics where the accepted definition looks more natural?  Are there some formulas that work out more cleanly with the accepted definition?