This is a curiosity from a reading a text that offered no proof. Why is $(C^{1}[0,1], \|\cdot\|_{\infty})$ not Banach?
The space $ C^1[0,1] $- the space of all continuously differentiable functions on $ [0,1]$ is not a Banach space with respect to the sup norm,$ \|.\|_{\infty} $ since the uniform limit of a continuously differentiable function need not be differentiable.  How can I illustate this statement using a counter example? Can I use $ f_{n}=\frac1 n \sin nx $ as a counter example?  Also, is $ C^1[0,1] $ is same as the space given by $ X=\{f\in C^1[0,1]:f(0)=0 \}$. Can I use the same example to show that this is not a Banach space?  More help is appreciated! Thanks!
What is the distinction between "among" and "amongst"? Insert only half a pdf page I'd like to include papers of mine which can easily be done with the pdfpages packages:  \usepackage{pdfpages} \includepdf[pages=-]{mypaper.pdf}   However, my thesis has an A5 format. Which means that each page is "half" an A4 page. Basically, I'd like two include each PDF page on two pages, like illustrated below:    While I found how to rotate the PDF using the angle parameter, I did not find how to split it... Determining a tricky limit What is the limit of   $$\lim_{n \to \infty} \frac{\log n}{n^\delta}$$  where $\delta &gt; 0$? When is $A- B = B- A$? Hello I am taking a Discrete Mathematics course and having some trouble with this question about sets:  Under what conditions is $A- B = B- A$  Diagram:   Diagram:   Maybe I'm understanding incorrectly, but how can A-B and B-A be equal if they contain elements that are not in each other? When I installed my SSD I just mounted with discard and didn't sweat it. However today I was reading about the pros and cons of using fstrim instead and decided to run the program to get an idea of how long it would actually take (still with my partitions mounted with discard). The command took several minutes on both my root and home partitions. For my home partition I used -v and got this:  $ sudo fstrim -v /home /home: 137494052864 bytes were trimmed   This is more than the amount of free space on the partition!  $ df -h /home Filesystem      Size  Used Avail Use% Mounted on /dev/sda2       206G   78G  118G  40% /home   Subsequent runs finish in less than a second, eg:  $ sudo fstrim -v /home /home: 0 bytes were trimmed   Surely if I have always had the partition mounted with discard, fstrim should not trim a large amount of data like that? The discard option is definitely enabled, here are the relevant fstab lines:  UUID=xxxxxxxx...    /          ext4   noatime,discard,errors=remount-ro  0      1 UUID=xxxxxxxx...    /home      ext4   noatime,discard,errors=remount-ro  0      2   And mount output lines:  /dev/disk/by-uuid/xxxxxxxx... on / type ext4 (rw,noatime,discard,errors=remount-ro,stripe=128,data=ordered) /dev/sda2 on /home type ext4 (rw,noatime,discard,errors=remount-ro,stripe=128,data=ordered)   The SSD is a TOSHIBA THNSNS256GMCP. Why does this happen? I'm travelling from Madrid to Brisbane, but I have a 9 hours layover at Dubai Airport. I do not plan to leave the airport, and the connection is with the same airline (Emirates). I'm from El Salvador.    Do I need a transit visa? Link broken on desktop but work on nautilus file manger I created a shortcut to a text file called &quot;imp&quot; on right-clicking it from the default file manager everything works well. But on clicking it on the desktop I get this  Any Idea why? How to toggle Scroll Lock in Excel 2010 without a Scroll Lock key in the keyboard I have a notebook with no Scroll Lock key (or any shortcut using the 'Fn' key).  Somehow Excel 2010 got the Scroll Lock on. I see it left down in the status bar and now I can't navigate through cells using the arrow keys, which is pretty annoying. It wasn't turned on some hours ago, so I think I must have hit some Scroll Lock toggle shortcut in Excel 2010 by mistake.  Anyone knows what hot to turn off Scroll Lock in Excel (shortcut or menu option)? Software center disappeared [19.10] I have had trouble with sound (now fixed) since upgrading to 19.10, and in the middle of trying the common solutions, the Software Center "disappeared". I got aware of quite late, so I don't know what caused it.  Anyway, would you know how to reinstall it ? Trying sudo apt-get install software-center doesn't work. Validation Error: Value is not valid How can I get a complete list of fields for a standard object? I am new to Apex, but it is apparent to me that most (if not all) standard objects probably have an ID field. However, this field does not appear in the reference for the standard objects in the Soap API Developer's Guide. Instead, it reads:     To verify the complete list of fields for an object, you can use a   describe call from the API, or inspect with an appropriate tool, for   example, inspecting the WSDL or using a schema viewer.   If I want to find out for sure whether an object has an ID field, or what the field is named (is it ID, Identifier, id, etc), it sounds like I need to use a "describe call" or "an appropriate tool" or inspect the "WSDL" using a "schema viewer".  Now, I am an experienced software developer, but those solutions are so ambiguous I have no idea where to look next to figure out how to get an object's ID. Can anybody either refer me to more specific documentation, or explain how to find the complete field documentation via a "describe call", "appropriate tool", or "schema viewer"? Which test should I use for ordinal data? In my experiment, I asked 50 participants to rank on a 5 point scale which voice they preferred. I summed each particpants' rating for each voice: the total for voice 1 is 95, the total for voice 2 is 186.  Am I correct in doing this? Or should I analyse the raw data?  Which test should I use for this data? Would a t-test work here? Alternatively I thought I could use a chi-square test.   I am interested in seeing if the preference for voice 2 is significant. Textbook on the Geometry of Special Relativity I am looking for a textbook that treats the subject of Special Relativity from a geometric point of view, i.e. a textbook that introduces the theory right from the start in terms of 4-vectors and Minkowski tensors, instead of the more traditional "beginners" approach. Would anyone have a recommendation for such a textbook ?  I already have decent knowledge of the physics and maths of both SR and GR ( including vector and tensor calculus ), but would like to take a step back and expand and broaden my intuition of the geometry underlying SR, as described by 4-vectors and tensors. What I do not need is another "and here is the formula for time dilation..." type of text, of which there are thousands out there, but something much more geometric and in-depth.  Thanks in advance. Html form in email Credit cards: How is a cash advance different from a purchase? Why are the fees so high? I've shown the existence of unique square roots of all positive rational numbers, so now I want to prove that the square root is monotone:   $0&lt;a&lt;b$ if and only if $\sqrt{a} &lt; \sqrt{b}$ Doing a search via regular Gmail frontend in the browser, shows strange results.  I want to search all mails from   *@ebay.*   My search:  from:@ebay   shows also emails sent from an ebay subdomain:   eBay@reply.ebay.com   It seems like Gmail ignores the "@", otherwise I can't see why those messages appear in the search results.  I also tried following searches with no success:  from:"@ebay" from:'@ebay' from:(@ebay) from:("@ebay") from:('@ebay') Massless charged particles Galois group of irreducible cubic equation The polynomial $f(x) = x^3 -3x + 1$  is irreducible, and I'm trying to find the splitting field of the polynomial.  We've been given the hint by our lecturer to allow an arbitrary $\alpha$ to be a root and then to check that $\alpha$ is a root, then $2-\alpha^2$ is also root. I've done this with no problem.  My issue is since this is a polynomial of degree $3$, there must be another root that is somehow related to $\alpha$ so that I can create the splitting field and in turn the Galois group of my polynomial but I have no idea what this last root could be. Is a strong background in maths a total requisite for ML? I'm starting to want to advance my own skillset and I've always been fascinated by machine learning. However, six years ago instead of pursuing this I decided to take a completely unrelated degree to computer science.   I have been developing software and applications for about 8-10 years now, so I have a good handle but I just can't seem to penetrate the maths side of machine learning/probabilities/statistics.   I start looking at learning material and on the first page it might include something which confuses me and immediately sets up a barrier in my learning.   Is a strong background in maths a total requisite for ML? Should I try and fill in the blanks of my maths before continuing with ML? Can self learning really work for just a developer without any hard computer science background?    Related question: