Let $F$ be a field. Then the Euclidean inner product on $F^n$ is defined as $\langle \mathbf{x},\mathbf{y}\rangle=x_1\overline{y_1}+\cdots+x_n\overline{y_n}$ for all $\mathbf{x}=(x_1,~x_2,~\cdots,~x_n),~\mathbf{y}=(y_1,~y_2,~\cdots,~y_n)\in F^n$. Then its induced norm, $\lVert x\rVert:=\sqrt{\langle\mathbf{x}, \mathbf{x}\rangle}$, is a $\ell^2$-norm on $F^n$. From this, I come up with a doubt that can it be possible that we define an inner product on $F^n$ such that its induced norm is $\ell^1$-norm, or $\ell^{\infty}$-norm, or even $\ell^{p}$-norm?
Please help me prove this. I'm not sure how to apply the parallelogram law to the norm.
Should I submit bank statements when applying for a UK Visa? What do they say about me? The UK Visas and Immigration Directorate publishes  that provides helpful tips that can result in successful visa applications. Section 2 of this guidance opens is entitled &quot;other documents you may want to provide&quot; (emphasis mine).  Section 2: other documents you may want to provide – all visitors This section provides guidance on the types of documents that you may want to provide to help us consider your application against the Immigration Rules. Previous travel documents/passports, which show previous travel. Financial documents showing that you have sufficient funds available.  These must clearly show that you have access to the funds, such as:  bank statements building society book proof of earnings such as a letter from employer confirming employment details (start date of employment, salary, role, company contact details) where a third party (who is either in the UK or will be legally in the UK at the time of your visit) is providing financial support to you e.g. a business, a friend or a relative, documents to show that they have sufficient resources to support you in addition to themselves and any dependant family should be provided   The first item is &quot;bank statements&quot; and this ties in with the , which says...  evidence that you can support yourself during your trip, eg bank statements or payslips from the last 6 months  (Note 'eg' is a Latin abbreviation meaning &quot;for example&quot;; this appears to be optional rather than a fixed requirement.) Question: How should someone interpret this such that the chances of success are maximized? How many statements should I submit? What are they looking for and what do my bank statements tell about me? I think this is a duplicate of an old question without a good answer:  My child was doing schoolwork on her school-issued Macbook Air. Suddenly lighter greys are totally white. This generally gives the images on the screen a vague feeling of being &quot;off&quot;, and pixellates all curved edges. The biggest problem is that text on some of her sites is unreadable. I've checked the following:  Color profiles - it looked like it was using the default, but a cycled through a few built-ins. I could see noticeable changes to color balance, but the light greys remained the same as white. Also deleted contents of ~/Library/ColorSync/. Accessibility - my child has accidentally inverted colors before, this isn't that. No other accessibility settings seemed related to color. Display settings - Modifying brightness was also fruitless. The screen overall got brighter or darker, but still couldn't distinguish between light grey and white Rebooting - colors look fine until shortly after logging in - seems connected to user profile somehow.  Here are some images to demonstrate the issue. This is a generic color test image:  On the Mac, it looks like this (simulated with photoshop):  Any solution would be appreciated, but if there is a keyboard shortcut that can cause this, it would be the most likely explanation (8 year old says they &quot;might have rested their arms on the keyboard&quot; during a class meeting) Full width image in beamer? Encountering a FileSystemRights value that isn't defined in enumeration Proof of the equality $\sum\limits_{k=1}^{\infty} \frac{k^2}{2^k} = 6$ Show that for $k$ running over positive integers  $$ \sum_{k=1}^\infty \frac{k^2}{2^k}=6 .$$ We can use finite calculus. Why is it differential equations exist on an interval instead of a domain? I understand a domain is the set of input elements a function is defined for (and can have breaks in it e.g. union of 2 sets) and a interval is a continuous range of real numbers. Why do we speak of the solution to a differential equation over an interval instead of an domain? If the solution to an ODE is just a function, and we normally speak of functions as having a domain, then why wouldn't we use domain? Do cards with abilities that target trigger heroic? Flat quads (or, why is my mesh so funky?) Let he who believes in this prophet speak now what he knows.       Let him who believes in this prophet speak now what he knows. What to do when witches appear early in the game? Why are my Cycles settings missing? so I'm working on a little bathroom scene and wanted to make a clay render. I started looking for the material override which I kinda knew where it was and on the way I switched my render engine accidently to workbench instead of cycles. Switched to cycles all my seetings on the render tab are missing except color management and free style. Hows that? I restarted blender multiple times and tried some other stuff but nothing changed a thing, and I'd rather avoid a new installation Resizing or expanding a CoreStorage volume Email from IOS with attachments larger than 7.5Mb gets stuck in outbox and constantly tries to resend. Costs us hundreds in data fees How am I supposed to use nvidia-prime? I have an acer laptop with these GPUs: - GPU0: Intel HD Graphics 4000 (M) (obviously integrated) - GPU1: NVIDIA GeForce GT 710M  I uninstalled all nvidia-drivers because I was experiencing a lot of bugs and I read here and there that they are not supposed to work with Optimus without either bumblebee or nvidia-prime.  As far as I understand, nvidia-prime would allow me to have the NVidia GPU always on. And to choose between that or always off (i.e. always use Intel)  So now I installed nvidia-prime with  sudo apt-get install nvidia-prime   What am I supposed to do now?  I thought nvidia-prime provided a graphical interface which would allow me to choose which graphic card to use, and even to install the needed nvidia drivers.  By typing "nvidia" in the dash, nothing shows up. From a terminal, the only command available starting with "nvidia" is "nvidia-detect" which just prints "none".  So, I installed nvidia-prime, how do I use it? I own a Windows 10 K or N Edition which doesn't include the media player package. My problem is: I want to have preview thumbnails in the explorer for my MP3s and videos.  I had the issue before when I updated to Win10 but there  saved me. Since the Anniversary update they are gone again.   Now I can install the old update but I still don't get the media player nor the preview images.  I tried most of the things Microsoft offers me but they are either not installable or don't work.  Any ideas what might be the correct update package?  Thx!  Ron Why can't energy be created or destroyed? Grub not loading after Windows 8 Install My system was configured to dual boot Ubuntu 12.04.1 LTS and Windows 7. Today I got my hands on the MSDN release of Windows 8 and I installed it over my Windows 7. Now the computer just boots to Windows 8 directly without loading the GRUB screen.  So I followed the steps as suggested in: . Running this command:      ls -l /dev/disk/by-label/   gives the following output:      total 0     lrwxrwxrwx 1 root root 10 Sep 11 07:51 Entertainment -&gt; ../../sda2     lrwxrwxrwx 1 root root 10 Sep 11 02:45 PENDRIVE -&gt; ../../sdb1   Also fdisk -l command gives this as the output:      Disk /dev/sda: 320.1 GB, 320072933376 bytes     255 heads, 63 sectors/track, 38913 cylinders, total 625142448 sectors     Units = sectors of 1 * 512 = 512 bytes     Sector size (logical/physical): 512 bytes / 512 bytes     I/O size (minimum/optimal): 512 bytes / 512 bytes     Disk identifier: 0x1246aa23         Device Boot      Start         End      Blocks   Id  System     /dev/sda1   *        2048   319582199   159790076    7  HPFS/NTFS/exFAT     /dev/sda2       319582208   602906623   141662208    7  HPFS/NTFS/exFAT     /dev/sda3       602908672   625135615    11113472   83  Linux      Disk /dev/sdb: 1939 MB, 1939865600 bytes     64 heads, 63 sectors/track, 939 cylinders, total 3788800 sectors     Units = sectors of 1 * 512 = 512 bytes     Sector size (logical/physical): 512 bytes / 512 bytes     I/O size (minimum/optimal): 512 bytes / 512 bytes     Disk identifier: 0xc3072e18         Device Boot      Start         End      Blocks   Id  System     /dev/sdb1   *        2248     3788799     1893276    c  W95 FAT32 (LBA)   So I assume that I have to run this: sudo grub-install /dev/sda3 to get GRUB up and running. But I am getting this error:      /usr/sbin/grub-probe: error: cannot find a device for /boot/grub (is /dev mounted?).   Can anyone please guide me in the right direction? The current Ubuntu installation is far too customized to my needs to lose it to a boot manager issue! Any help is much appreciated! Ugly 'post date' alignment on user profile when the date wraps into multiple lines Show that the operator norm is submultiplicative We had in our lecture on numerical analysis the following: Let $\mathrm{Lin}(X,Y)$ be the set of all linear maps $X\rightarrow Y$. Let $A\in\mathrm{Lin}(\mathbb R^l,\mathbb R^n)$ and $B\in\mathrm{Lin}(\mathbb R^n,\mathbb R^m)$ and $\|C\|_{op}:=\max_{\|x\|\leq1}\|C(x)\|$. Then our lecturer followed $\|A\circ B\|_{op}\leq\|A\|_{op}\cdot\|B\|_{op}$. So he didn't prove it and so I've tried it by my own. My attempt: $$\begin{aligned} \|A\circ B\|_{\mathrm{op}} &amp;= \max_{\|x\| &lt; 1}\|(A \circ B)(x)\| \\ &amp; \leq \max_{\|x\| &lt; 1} \|A\|_{\mathrm{op}} \|B(x)\| \\ &amp;= \|A\|_{\mathrm{op}} \max_{\|x\|\leq1}\|B(x)\|\\ &amp;= \|A\|_{\mathrm{op}}\|B\|_{\mathrm{op}} \end{aligned}$$ But this seems too easy. I am really interested in a nice proof so anybody could help? Thanks a lot! How can I prevent inline math formulas from overflowing into the margin? When I compile my document in LaTeX, some formulas in the document in math mode, between $ $, go into the right margin area beyond all the text.  Is there a way to fix this aesthetic problem?