I know there are standard proofs for this theorem, but I need to prove it by contradiction or proving that $\dim X=\infty$.  I thought maybe using Hahn-Banach?  Thanks.
Can you read my proof and tell me if it's correct? Thanks.  Let $V$ be a vector space over a complete topological field say $\mathbb R$ (or $\mathbb C$) with $\dim(V) = n$, base $e_i$ and norm $\|\cdot\|$. Let $v_k$ be a Cauchy sequence w.r.t. $\|\cdot\|$. Since any two norms on a finite dimensional space are equivalent, $\|\cdot\|$ is equivalent to the $l^1$-norm $\|\cdot\|_1$ which means that for some constant $C$, $\varepsilon &gt; 0$, $k,j$ large enough, $$ \varepsilon &gt; \|v_j - v_k\| \geq C \|v_j - v_k\|_1 e_i= C \sum_{i=1}^n |v_{ji} - v_{ki}| \geq |v_{ji} - v_{ki}|$$ for each $1 \leq i \leq n$. Hence $v_{ki}$ is a Cauchy sequence in $\mathbb R$ (or $\mathbb C$) for each $i$. $\mathbb R$ (or $\mathbb C$) is complete hence $v_i = \lim_{k \to \infty} v_{ki} $ is in $\mathbb R$ (or $\Bbb C$) for each $i$. Let $v = (v_1, \dots , v_n) = \sum_i v_i e_i$. Then $v$ is in $V$ and $\|v_k - v\| \to 0$:  Let $\varepsilon &gt; 0$. Then $$ \|v_k - v\| \leq C \|v_k - v\|_1 = C \sum_{i=1}^n |v_{ki} - v_i| \leq C^{'}n \varepsilon$$  for $k$ large enough.
When I used  pip install ipython   or  pip install "ipython[notebook]"   I get the following error:  Downloading/unpacking ipython   Downloading ipython-2.3.1-py27-none-any.whl (2.8MB): 2.8MB downloaded Installing collected packages: ipython Cleaning up... Exception: Traceback (most recent call last):   File "/usr/lib/python2.7/dist-packages/pip/basecommand.py", line 122, in main     status = self.run(options, args)   File "/usr/lib/python2.7/dist-packages/pip/commands/install.py", line 283, in run     requirement_set.install(install_options, global_options, root=options.root_path)   File "/usr/lib/python2.7/dist-packages/pip/req.py", line 1435, in install     requirement.install(install_options, global_options, *args, **kwargs)   File "/usr/lib/python2.7/dist-packages/pip/req.py", line 671, in install     self.move_wheel_files(self.source_dir, root=root)   File "/usr/lib/python2.7/dist-packages/pip/req.py", line 901, in move_wheel_files     pycompile=self.pycompile,   File "/usr/lib/python2.7/dist-packages/pip/wheel.py", line 206, in move_wheel_files     clobber(source, lib_dir, True)   File "/usr/lib/python2.7/dist-packages/pip/wheel.py", line 193, in clobber     os.makedirs(destsubdir)   File "/usr/lib/python2.7/os.py", line 157, in makedirs     mkdir(name, mode) OSError: [Errno 13] Permission denied: '/usr/local/lib/python2.7/dist-packages/IPython'  Storing debug log for failure in /home/username/.pip/pip.log   What does this error mean? How do I fix it? Can I buy (liquid) duty free at Dublin airport, before going through pre-clearance? Following on from my , I've been watching the  from the Dublin airport website. That mentions that after checkin, passengers clear normal Irish security checks, then proceed (via optional shopping) to a TSA-style security check before reaching the CBP juxtaposed border control. It also mentions that duty free shopping is available after immigration and customs.  That leads me to wonder - can I buy duty free liquids in the main bit of Dublin Airport, before reaching the US pre-clearance area? Or must all duty free liquids (eg alcohol) be purchased in the smaller duty-free shop after US pre-clearance just before boarding? Sam developed an annual sweepstakes promotion entitled “The Free Gas Punt Return Game”.    Do I need a comma after entitled or not? How to find a Java Memory Leak How do you find a memory leak in Java (using, for example, JHat)? I have tried to load the heap dump up in JHat to take a basic look. However, I do not understand how I am supposed to be able to find the root reference () or whatever it is called. Basically, I can tell that there are several hundred megabytes of hash table entries ([java.util.HashMap$Entry or something like that), but maps are used all over the place... Is there some way to search for large maps, or perhaps find general roots of large object trees?  [Edit] Ok, I've read the answers so far but let's just say I am a cheap bastard (meaning I am more interested in learning how to use JHat than to pay for JProfiler). Also, JHat is always available since it is part of the JDK. Unless of course there is no way with JHat but brute force, but I can't believe that can be the case.  Also, I do not think I will be able to actually modify (adding logging of all map sizes) and run it for long enough for me to notice the leak. In FreeBSD and also in Linux, how can I get the numerical chmod value of a file? For example, 644 instead of -rw-r--r--? I need an automatic way for a Bash script. Alt-Tab shows two application switchers on Unity I started to have problems with my Ubuntu application switcher (Alt +  Tab) this morning. When I press Alt +  Tab, not only one, but two application switchers show up. Last night I customized some stuff using the Unity Tweak Tool. I thought I didn't touch the application switcher. So now it looks like this: Suppose I installed ubuntu with the user usr1 then I added usr2. I then create a group grp1 where this 2 users are added. A folder is owned by usr1 and has group grp1. But when I login as usr2 I cannot write. The folder has permissions 755...  usr2@kmserver:/media/data/www$ groups usr1 usr1 : grp1 ... usr2@kmserver:/media/data/www$ groups usr2 usr2 : grp1 ... usr2@kmserver:/media/data$ ls -l total 20 drwx------ 2 root root 16384 2011-05-12 10:53 lost+found drwxrwxr-x 2 grp1   grp1    4096 2011-05-12 15:15 www   Whats wrong? I have two Date values. I need to find number of weeks between them. I want to do it Apex Class. Not looking for formula fields.  So far I tried:  Integer numberofweeks = (projectStartDate - opp.Project_Start_Date__c)/7;    I receive this error:     Date arithmetic expressions must use Integer or Long arguments remote action function encountering CPU time out limit exception CPU time out limit was being hit while processing child records on the inline opportunity visualforce page, to handle this i wrote a remoteActionFunction on the page that invokes controller method that's taking long to process and when i get CPU LIMIT exception in the result, i am invoking javascript call back function that invokes a action function which handles the same code asynchronously(@future).  Question: by setting the timeout (Visualforce.remoting.timeout = 120000;) can't we overcome CPU TIME out limit of 10,000 ms. block certain dangerous commands? I have a CSV containing x,y, and z coordinates. The CSV also has a fourth column containing information on sphere diameter. I need to create a sphere at each coordinate, having the corresponding diameter. Can someone provide me a basic Python script that performs this task? My actual task is much more involved, but I think I can solve it if I have the basics in place. I have an element that already has a class: &lt;div class=&quot;someclass&quot;&gt;     &lt;img ... id=&quot;image1&quot; name=&quot;image1&quot; /&gt; &lt;/div&gt;  Now, I want to create a JavaScript function that will add a class to the div (not replace, but add). How can I do that? I can't install Rainlendar 2 on my Ubuntu 18.04, I found this solution for 16.04 (), followed it, seems to work, but after install, there's no Rainlendar in my Show Apps, so I'm not sure what went wrong. When I issued sudo dpkg -i rainlendar2-pro_2.12.2.b138-1_amd64.deb, I got this:   Selecting previously unselected package rainlendar2-pro. (Reading database ... 190677 files and directories currently installed.) Preparing to unpack rainlendar2-pro_2.12.2.b138-1_amd64.deb ... Unpacking rainlendar2-pro (2.12.2.b138-1) ... dpkg: dependency problems prevent configuration of rainlendar2-pro:  rainlendar2-pro depends on libpng12-0; however:   Package libpng12-0 is not installed.  rainlendar2-pro depends on tofrodos; however:   Package tofrodos is not installed.  rainlendar2-pro depends on libjpeg62; however:   Package libjpeg62 is not installed.  rainlendar2-pro depends on libwebkitgtk-1.0-0; however:   Package libwebkitgtk-1.0-0 is not installed.  dpkg: error processing package rainlendar2-pro (--install):  dependency problems - leaving unconfigured Processing triggers for gnome-menus (3.13.3-11ubuntu1) ... Processing triggers for desktop-file-utils (0.23-1ubuntu3.18.04.1) ... Processing triggers for mime-support (3.60ubuntu1) ... Errors were encountered while processing:  rainlendar2-pro   I got this after issuing this command in terminal sudo apt-get -f install  Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages will be REMOVED:   rainlendar2-pro 0 upgraded, 0 newly installed, 1 to remove and 35 not upgraded. 1 not fully installed or removed. After this operation, 20.5 MB disk space will be freed. Do you want to continue? [Y/n] y (Reading database ... 191316 files and directories currently installed.) Removing rainlendar2-pro (2.12.2.b138-1) ... Processing triggers for mime-support (3.60ubuntu1) ... Processing triggers for desktop-file-utils (0.23-1ubuntu3.18.04.1) ... Processing triggers for gnome-menus (3.13.3-11ubuntu1) ...   What I'm doing wrong? After the last command it just removed Rainlendar and not installing dependencies as stated in the original post? TIA How do you create grass? How do I get back some deleted records? I mistakenly deleted around 2,000,000 records from a remote SQL Server 2008 table.  The server is not granting me access to the backup files on the server side.   Is there any way to get back these records? Why does exclamation mark within double quotes cause a Bash error? Please look at these commands:  $ notify-send SYNC TIME! $ notify-send 'SYNC TIME!' $ notify-send "SYNC TIME!" bash: !": event not found $   The first two commands produce a notification bubble as expected. The third gives the error shown.  and  $ echo SYNC TIME! SYNC TIME! $ echo 'SYNC TIME!' SYNC TIME! $ echo "SYNC TIME!" bash: !": event not found $   Here as well, the echo works for first two commands but not in the third.  More problems here (although I was not planning on using this): both notify-send "SYNC!TIME" and echo "SYNC!TIME" give bash: !TIME": event not found.  But both notify-send and echo work with "SYNC! TIME"  Can someone please explain why the bash: !": event not found error appears? Inertial frames of reference I'm struggling with the notion of an inertial frame of reference. I suspect my difficulty lies with the difference between Newtonian and relativistic inertial frames, but I can't see it.   I've read that Newton's laws apply in any non accelerating frame of reference, which are called inertial frames. So, if I play pool on a train moving with uniform velocity, the balls behave in the same way as if I were playing pool in a pool hall. So the train is (to a good approximation, ignoring forces caused by the Earth's rotation and movement around the Sun etc) an inertial frame.   I've also read that in an inertial frame, Newton's first law is satisfied. So, if I slide a rock on a sheet of ice and I could somehow eliminate the frictional forces between the rock and the ice, the rock would carry on sliding for ever, as predicted by Newton's first law.   Question 1 - is the  sheet of ice also therefore a good approximation to an inertial frame?  Question 2 - are these two definitions of inertial frames saying the same thing in different ways?  Both these examples occur in gravitation fields, which doesn't seem to matter as both the train and the ice sheet are presumably good approximations to inertial frames.  Question 3 - is gravity irrelevant when defining these two inertial frames?  In special relativity, I've read (Foster and Nightingale) that an inertial frame is also one where Newton's first law holds. But as it's special relativity there cannot be an inertial frame if there's a gravitational field, so I'm assuming the above two examples are not inertial frames in special relativity.  Question 4 - how come you can use the same definition of an inertial frame (satisfied Newton's first law) but in the case of special relativity, the train and the ice sheet aren't inertial frames. Is this to do with not being able to synchronize clocks in a gravitational field?  In general relativity, I've read (Schutz) that a freely falling frame is (on Earth) the only possible (and local) inertial frame. So again the first two examples would not be inertial frames in general relativity.  Question 5 - am I right to think that my two examples aren't even approximate approximations to an inertial frame in relativity?  Question 6 - have I missed anything else that might be useful?  Many thanks  Edit. On reflection, am I right in thinking that because Newtonian mechanics assumes universal, absolute time we don't need to worry about synchronizing clocks in a Newtonian inertial frame. Therefore we don't need to worry about gravity in a Newtonian inertial frame, because in such a frame gravity does not affect time.  This is not the case in spacetime, as here gravity does affect time and the only way to have synchronized clocks in an inertial frame in spacetime is: 1. have no gravitational field, or 2. use a local, freely falling frame.  Am I on the right track here? Installing gcc-7 (and newer packages) in Ubuntu 16.04 'help' link is located lower than the other items It shows like this on Firefox 25 and Internet Explorer 11 (minus the red line of course):    Chrome seems to be fine. Finite ring has only zero divisors and units If a nonzero element $a$ of a ring $\mathbb{D}$ does not have a multiplicative inverse, then $a$ must be a zero divisor.   If $\mathbb{D}$ has finitely many elements, then the statement is true. However, I don't see why, as for the infinite case, 7 isn't a zero divisor only because every element $a &gt; 0$ when multiplied by a $b &gt; 0$ gives you $ab &gt; 0$ which has nothing to do with infinity or finite amounts of elements.