Given an infinite-dimensional normed vector space M, there exists a linear functional $ f: M \rightarrow K $, where $K$ is the filed of scalars.   I am trying to prove it following the strategy of dealing with finite dimensional basis. Can somebody say how one should procede ?  Thanks.
I'm trying to find a discontinuous linear functional into $\mathbb{R}$ as a prep question for a test. I know that I need an infinite-dimensional Vector Space. Since $\ell_2$ is infinite-dimensional, there must exist a linear functional from $\ell_2$ into $\mathbb{R}$. However, I'm having trouble actually coming up with it.   I believe I'm supposed to find an unbounded function (although I'm not sure why an unbounded function is necessarily not continuous; some light in that regard would be appreciated too), so I thought of using the vectors $e^i$, which have all entries equal to zero, except for the $i$-th one. Then, you can define $f(e^i)=i$.  That'd be unbounded, but I'm not sure if it'd be linear, and even if it is, I'm not sure how to define it for all the other vectors in $\ell_2$.   A friend mentioned that at some point the question of whether the set $E=\{e^i:i\in\mathbb{Z}^+\}$ is a basis would come up, but I'm not sure what a basis has to do with continuity of $f$.   I'm just learning this topic for the first time, so bear with me please.   The space of sequences that are eventually zero (suggested by a few people) turned out to be exactly what I needed.  It also helped to cement the notions of Hamel basis, not continuous, etc.
How do I convince someone that $1+1=2$ may not necessarily be true? Me and my friend were arguing over this &quot;fact&quot; that we all know and hold dear. However, I do know that $1+1=2$ is an axiom. That is why I beg to differ. Neither of us have the required mathematical knowledge to convince each other. And that is why, we decided to turn to Math Stackexchange for help. What would be stack's opinion? Here is what it looks like in Blender Render (Not Cycles).   This is how I set it up.   Basically, I'm trying to export it, but it's not including the &quot;Freestyle&quot; lines. When I open up the .dae file in Finder, it looks like this:   What is wrong here? I want it to be exported (COLLIDA .dae) with the black outline. I'm seeming to think that maybe the &quot;Freestyle&quot; is only for the Blender Render. Am I right? How would I go about getting the black outline included in my exported model? Is it an export setting I have to tick? Thanks for any helpful answers. How do you convert a C++ string to an int? Possible Duplicate:          How do you convert a C++ string to an int?  Assume you are expecting the string to have actual numbers in it ("1", "345", "38944", for example).  Also, let's assume you don't have boost, and you really want to do it the C++ way, not the crufty old C way. How to find continued fraction of pi I have always been amazed by the continued fractions for $\pi$. For example some continued fractions for pi are:  $\pi=[3:7,15,1,292,.....]$ and many others given .  Similarly some nice continued fractions for $e$ and it's derivatives are given .I have tried to prove that the are indeed the continued fractions but did not get very far.If any one can can help it would be great(especially if there is some easy way to get continued fraction of derivatives of e from the original continued fraction of e).Here derivatives of e I mean $\sqrt e,\frac{e-1}{e+1} $ etc. How to create batch file in Windows using "start" with a path and command with spaces I need to create a batch file which starts multiple console applications in a Windows .cmd file.  This can be done using the start command.  However, the command has a path in it.  I also need to pass paramaters which have spaces as well.  How to do this?  E.g. batch file  start "c:\path with spaces\app.exe" param1 "param with spaces" How to make a WMS GetFeatureInfo request in Leaflet? I am trying to make a WMS GetFeatureInfo request using Leaflet. I can not find something in the documentation to describe this process. Can you please provide with an example demostraing how to do something like this? What changes when an Alienist becomes an outsider? Omitting "is", like in "I think it strange" What is the grammar behind the construction "I think it strange/necessary that ...", and when can and cannot this apparent omission of the copula be used? Do we always need the "that" clause?  Also, this seems to be only allowed when the object is a pronoun (it in the above example). That is, we don't say "I think this method strange", correct? If a SHA256 hash with high entropy is then hashed with one made from low entropy, is the resulting hash higher/same/lower entropy? If one were to create a SHA256 hash using 256 coin flips converted to hex (which, as I understand it, is as "entro-phized" as one can capture in SHA256) then take that hash as a string, combine that with a hash of the word "cat" and hash the resulting 512 character string, would the resulting hash from that exercise be any less random than our "perfect" hash we began with?  My thinking is the final string would capture the same amount as the original randomness as what we started with because the "extra" entropy can not be captured and is lost. Is this the case and am I thinking about this correctly? Would hashing something with high entropy with something simpler ever cause a lower amount of randomness or does it only go up?  Thanks to anyone who can help explain this a little better. Negative integer division surprising result Setting the hostname: FQDN or short name? I've noticed that the "preferred" method of setting the system hostname is fundamentally different between Red Hat/CentOS and Debian/Ubuntu systems.   and the  say the hostname should be the FQDN:     HOSTNAME=&lt;value&gt;, where &lt;value&gt; should be the Fully Qualified Domain   Name (FQDN), such as hostname.example.com, but can be whatever   hostname is necessary.   The  is slightly more ambiguous:     Setup prompts you to supply a host name for this computer, either as a   fully-qualified domain name (FQDN) in the format hostname.domainname   or as a short host name in the format hostname.    says the hostname should not use the FQDN:     3.5.5. The hostname      The kernel maintains the system hostname. The init script in runlevel   S which is symlinked to "/etc/init.d/hostname.sh" sets the system   hostname at boot time (using the hostname command) to the name stored   in "/etc/hostname". This file should contain only the system hostname,   not a fully qualified domain name.   I haven't seen any specific recommendations from IBM about which to use, but  seems to have a preference.  My questions:   In a heterogeneous environment, is it better to use the vendor recommendation, or choose one and be consistent across all hosts? What software have you encountered which is sensitive to whether the hostname is set to the FQDN or short name? How do I ensure that a Python while-loop takes a particular amount of time to run? I recently upgraded Windows 7 to Windows 10. The biggest issue I have now is that the Windows Search service no longer works, making the Start menu mostly unusable.  If I go to Indexing Options in the control panel I see that Indexing is not running and the list of paths to index is empty.  Going to advanced options, the current index path is empty, and the new path is empty too. This is true even if I try to use the "New Index" button.  Using the troubleshooter won't help since the service doesn't start. If I try to manually start the "Windows Search" service, I get the error  "Error 3: Unable to find specified path" (this is the rough translation, the original message is in my language).  What can I do to find out which path/file is missing ? I tried to check with Process Monitor, but the only file (beside system libraries) the SearchIndexer.exe try to access seem to be c:\ProgramData\Microsoft\Search\Data\Application (everytime the service is started the whole folder c:\ProgramData\Microsoft\Search is deleted and recreated) and C:\Windows\System32\config\systemprofile (which exists).  Edit: all available updates are installed, and issuing      SFC /scannow   Doesn't reveal any issue Why should the interface for a Java class be preferred? would report a violation for:  ArrayList&lt;Object&gt; list = new ArrayList&lt;Object&gt;();   The violation was "Avoid using implementation types like 'ArrayList'; use the interface instead".  The following line would correct the violation:  List&lt;Object&gt; list = new ArrayList&lt;Object&gt;();   Why should the latter with List be used instead of ArrayList? Applications of complex numbers to solve non-complex problems Recently I asked a question regarding the diophantine equation $x^2+y^2=z^n$ for $x, y, z, n \in \mathbb{N}$, which to my surprise was answered with the help complex numbers. I find it fascinating that for a question which only concerns integers, and whose answers can only be integers, such an elegant solution comes from the seemingly unrelated complex numbers - looking only at the question and solution one would never suspect that complex numbers were lurking behind the curtain!  Can anyone give some more examples where a problem which seems to deal entirely with real numbers can be solved using complex numbers behind the scenes? One other example which springs to mind for me is solving a homogeneous second order differential equation whose coefficients form a quadratic with complex roots, which in some cases gives real solutions for real coefficients but requires complex arithmetic to calculate.  (If anyone is interested, the original question I asked can be found here: )  EDIT:  I just wanted to thank everyone for all the great answers! I'm working my way through all of them, although some are beyond me for now! Way to detect frozen video with ffmpeg? Is there a way to detect frozen video with ffmpeg? I only need to know if there are rather large regions of frozen video, say 10 seconds or more.   I'm familar with the  filter.   Maybe you could first difference adjacent frames, then use blackdetect to tell whether the difference has not changed? I'm not sure how to perform the difference though. Let $p$,$\ q$ be prime numbers and $n\in \mathbb{N}$ such that $p \nmid n-1$. If $p \mid n^q-1$, then $q \mid p-1$. How does the momentum operator act on state kets? I develop an application in which i constantly get samples of heart pulse.  I defined an interval of t seconds.  In each t seconds I have n samples.  In every interval, I want to calculate the tendency of those n samples. For example, lets say I have n = 5, and I have samples with values {70, 88, 95, 103, 115}. I want to recognize I have a growth in the heart pulse, and I want to have some measure for the rate of grows/decrease/almost no change.  I thought on two approaches for solving this problem.   I calculate the linear approximation for the n samples using linear regression by applying least squares implemented by the normal equations. (I treat each sample as two coordinates vector with x coordinate as time and y coordinate as heart pulse). I get from the normal equations a linear function of the form y = mx+b and my measure for the tendency is the slope, i.e. the m value. The second approach is to calculate the correlation (pearson's correlation) between vector x and vector y when x is the time and y is the heart pulse.   I'm asking which approach do you think is better for my problem (determining the tendency of heart pulses). Or, do you have a better algorithm for solving this problem? “between” vs. “among” Today I was cut off in the middle of the following sentence:     Between Cook, Strauss, and Pietersen—   My friend said I was wrong. He said that for more than two entities, among/amongst are used, and that between is only for two entities.   I vaguely remember some rule like this, but I asked him to hear the whole sentence, which was:     Between Cook, Strauss, and Pietersen, they’ve notched up 56 centuries in Test cricket.   Should I have used among here? Seems correct to me to use between, but my friend was certain.   Another friend pointed out the following sentence:     I’m still choosing between Harvard, Yale, and MIT.    Certainly among doesn’t seem to fit here, and she, to me, was correct in using between.