Subset of separable Hilbert space I have a separable Hilbert space H. How can one prove that the closed unit ball B is separable?  I'm told it's trivial but I can't see it.  My initial idea is to take the set A (dense set of H) and intersect it with B. This new set is clearly constable and it's closure is in B. but how can I show that B is in th closure of this set?  Thanks for any help
Subspaces of separable normed spaces Let $X$ be a separable normed space. Is it true that every subspace is separable? If it was Hilbert space I would take the dense set and then their projections.  It sounds trivial but I cannot prove or disprove it...  Thank you.
What to do after visiting unsafe site If light rays obey to the wave equation, why can they be thought as straight lines? I'm a newbie with physics but I'm wondering how a ray of light can essentially be represented. I have always known that a ray of light proceeds in a straight line until it encounters another object (or material) that refracts/reflects it. But a light ray should be part of an electro-magnetic wave, is this correct? If it is so, it should obey to the  and this doesn't seem to me to describe a straight line ray.  I'm having problems visualizing how light is emitted and how it relates with the wave equation. Can someone with a clear understanding of the problem explain it to me in simple terms? What is necessary and/or sufficient for polynomials to provide isomorphic quotient rings? Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$":  $$\begin{array}{|c|c|c|} \hline p&amp;q&amp;p\implies q\\ \hline T&amp;T&amp;T\\ T&amp;F&amp;F\\ F&amp;T&amp;T\\ F&amp;F&amp;T\\\hline \end{array}$$  My understanding is that "$p\implies q$" means "when there is $p$, there is q". The second row in the truth table where $p$ is true and $q$ is false would then contradict "$p\implies q$" because there is no $q$ when $p$ is present.  Why then, does the third row of the truth table not contradict "$p\implies q$"? If $q$ is true when $p$ is false, then $p$ is not a condition of $q$.  I have not taken any logic class so please explain it in layman's terms.       Administrative note. You may experience being directed here even though your question was actually about line 4 of the truth table instead. In that case, see the companion question  And even if your original worry was about line 3, it might be useful to skim the other question anyway; many of the answers to either question attempt to explain both lines. Acro package provides "! Missing number, treated as zero." after update to acro-v2.7b What is the relationship between size of object with distance? How does the size/length of an object vary with distance?   Is it a logarithmic relationship? exponential? linear?   I plotted a curve of the size/length of an object for different distances from the camera, and the curve looked exponential/logarithmic. I was trying to understand the reasoning behind that. Multiple Apple Developer Accounts Experimental Fractional scaling makes fonts dirty in Ubuntu 17.10 I updated to 17.10 from 17.04.   I activated the fractional scaling by running the following command.   gsettings set org.gnome.mutter experimental-features "['scale-monitor-framebuffer']"   Then, I changed 125% scale. The scale was changed, but the fonts and icons are so dirty.    The launcher icon and font in Google Chrome and other application except the system setting are dirty.  How can I resolve this problem?  For some reason, font in the system setting is not dirty... How to update libavcodec on ubuntu 14.04? On Ubuntu 14.04.5 I just upgraded firefox to version 50.0; now there is a message on some pages which reads:   libavcodec may be vulnerable or is not supported, and should be updated to play videos   So how can I update libavcodec?  A simple   sudo apt-get install libavcodec   gives the following output:  Reading package lists... Done Building dependency tree        Reading state information... Done E: Unable to locate package libavcodec   So maybe I don't have libavcodec installed in the first place? So why do I get this message...? How to partition an encrypted disk I've installed Ubuntu 13.10 and formatted my whole HDD ,and i chose to encrypt the disk and it automatically chose the LVM option (and put it as a logical partition) now after Ubuntu installed i tried to repartition the disk thru Live CD put GPartition gave me it's an encrypted partition and it can't be resized , i want to put the system's files and the home folder on partition and my private files on another one  Any help please ? Handling mouse events for tool called by button? I have a dockable window with a 2 buttons that call 2 different tools (Draw Marker and Select Features), What i want to happen is when the user clicks on the relevant button on the dockable window and then clicks on the map, a onMouseUp event is called and it runs code i need to run.   Where does the onMouseUp event need to be placed so this can be done? Is it possible for a system to have negative potential energy? With respect to both gravity and electromagnetism, to the best of my understanding potential energy is added or subtracted from a system based on the distance between two objects such as charged or massive objects.  Would this then imply that when oppositely charged objects or two massive objects are brought together within a small enough radius that the drop in potential energy would be greater than the combined mass of the two objects making the net mass/energy of the system negative? Expected prediction error - derivation Find the number of solutions of $x^2=x\mod m$ for any integer $m$.  I was thinking about bringing the $x$ over to the other side and somehow apply chinese remainder thm. Space war, humans recreated, grown from plants with an alien female supervisor This was a small hardback book I read in the UK in the late eighties. No memory of the cover.  Some kind of space war and a long lived alien female escapes the conflict in her starship, her side had lost.  An evil enemy that she needs to trap so she gets (I think) human  DNA from space dust and sets up on a habitable planet. The story then gets like episodic and looks at the world every couple of generations while she hides out there.  Plants bud on the world and people pop out from big seedpods, the population listens out for babies crying and rescue them.  There are legends of the 'Earth mother’ who secretly guides them to become high tech.   In reality she is creating attractive food bait for the bad aliens.  I think an early human and her are attracted to each other and she (maybe) gives him long life or (maybe) uses reincarnation to keep seeing him.  Edit:- the name 'Rimstalker' has just now popped up in my brain! Do I need to pay in full for hostel reservation to show proof of accommodation for a Schengen visa? I am applying for a Schengen visa to travel to Germany and one of the requirements is showing proof of accommodation. I booked all my hostels via hostelworld.com, but only paid the non-refundable deposit (a couple of dollars for each place) instead of the entire amount. I intend to pay the rest of it on arrival at the hostel (or cancel it and go CouchSurfing, thus saving money).  But for the visa application, will the non-refundable deposit be enough to show proof of accommodation?  The reason I ask is, I emailed one of the hostels requesting a document showing my name and accommodation confirmation, and they replied saying they could only give me such a document once I pay in FULL.  Is this how I'm supposed to book for all places?  Have I been doing it wrong all this while (by only paying the tiny non-refundable deposit instead of full hostel fare)? -> 683  -> 678  ???  (Doesn't really bother me too much but figured I'd report the bug...) I am using Log4Net with the AdoNetAppender to log messages from a simple systray application into a SQL Server 2005 database.  I want to log the machine name along with the log message because this application will be running on multiple machines and I need to know on which one the message originated.  But, I cannot find a way to expose this information via the log4net.Layout.PatternLayout that I am using with the appender.  Is there a way to log the machine name via log4net in this manner? How do I get a Seraph Crystal? I just learned that using CBC encryption with an IV which is predictable is not secure.  From what I understand, using certain plain texts, and then guessing the IV that it uses, the attacker can verify if the IV he guessed was right. How does this qualify as an attack?