"well-known" inequality for numerical radius of an operator Let $H$ be a Hilbert space. For $T\in\mathcal{B}(H)$, define the numerical radius of $T$ as $\omega(T):=\displaystyle{\sup_{\|x\|=1}|\langle Tx,x\rangle|}$. I am trying to prove that $\|T\|\leq2\omega(T)$ but I just don't see it. My text simply says that this is "easy" and all the literature online simply refer to this estimate as "well-known".   I know that, for self-adjoint operators, it is $\|T\|=\omega(T)$ and it can be proven elementary. I thought that maybe writing $T=X+iY$ with $X,Y$ self-adjoint could help, but then I couldn't estimate the numerical radii of $X,Y$ related to $\omega(T)$.
Upper bound for norm of Hilbert space operator It is a standard result that for a bounded self-adjoint operator $T$ on a complex Hilbert space $H$, we have $\|T\|=\sup_{\|x\|=1}|\langle Tx,x\rangle|:=M(T)$, where $M(T)$ is called the  of $T$. It seems that for any bounded operator on $H$, we have $||T||\leq 2M$. Why is that so?
Evaluating logistic regression and interpretation of Hosmer-Lemeshow Goodness of Fit How does some of the black hole (BH) mass escape the event horizon (EH) of either BH, or the merged EH of two merged BHs? Why is $\omega$ the smallest $\infty$? Is there any scrappable junk I should hold on to that could otherwise be used for other purposes? When I open the file in Vim, I see strange ^M characters.  Unfortunately, the world's favorite search engine does not do well with special characters in queries, so I'm asking here:   What is this ^M character? How could it have got there? How do I get rid of it? How to remove extra start item from boot menu? I bought a laptop recently. While Windows 7 was installing, the battery died. So I plugged in the charger and re-installed Windows 7 Home Basic from the start. After completion of the procedure I now have 2 Windows folders in my C:\ drive, so I deleted the old Windows folder.  The problem is, while booting time it shows two Windows 7 options, although I already deleted the old Windows folder.  How can I remove old Windows 7 option from my boot? I want to upgrade my Ubuntu 13.04 to Ubuntu 13.10. I'm unable to do via GUI.   [ The reason why I'm unable to do so:  Via software updater, I proceed ahead and click upgrade (for upgrading my OS). I authenticate by providing my password. Then it proceeds and downloads upgrade tool. That time only (while downloading the tool) error message comes "system program problem detected". It then doesn't proceed ahead and I click to report the problem ]  So, I tried via command line but here also I couldn't succeed upgrading my OS.  For the same I ran   $ sudo apt-get update   Then I ran  $ sudo apt-get dist-upgrade   And then:  ravbholua@ravbholua-Aspire-5315:~$ sudo do-release-upgrade Checking for a new Ubuntu release Get:1 Upgrade tool signature [198 B]                                                                                                  Get:2 Upgrade tool [1,135 kB]                                                                                                         Fetched 1,135 kB in 6s (32.9 kB/s)                                                                                                    authenticate 'saucy.tar.gz' against 'saucy.tar.gz.gpg'  gpg exited 2 Debug information:    gpg: Signature made Friday 01 November 2013 10:34:27 AM IST using DSA key ID 437D05B5 gpg: Can't check signature: public key not found  Authentication failed Authenticating the upgrade failed. There may be a problem with the network or with the server.  ravbholua@ravbholua-Aspire-5315:~$    Please suggest how to go ahead and upgrade my OS. Assuming a simple grep such as:  $ psa aux | grep someApp 1000     11634 51.2  0.1  32824  9112 pts/1    SN+  13:24   7:49 someApp   This provides much information, but as the first line of the ps command is missing there is no context for the info. I would prefer that the first line of ps be shown as well:  $ psa aux | someMagic someApp USER       PID %CPU %MEM    VSZ   RSS TTY      STAT START   TIME COMMAND 1000     11634 51.2  0.1  32824  9112 pts/1    SN+  13:24   7:49 someApp   Of course, I could add a regex to grep specifically for ps:  $ ps aux | grep -E "COMMAND|someApp"   However, I would prefer a more general solution as there are other cases in which I would like to have the first line as well.  Seems like this would be a good use case for . How to create this curved shadow in Photoshop? How to create the shadow effect like the URL/image below? Specifically, the rounded/curved shadow at the bottom: I have an uploaded video and I want to cut sections of audio (strong language) or add beeps on top without reuploading the video. What is the equivalent of Persian idiom "When the reed blooms"? In Persian, we say "When the  blooms" when we want to  express that an event:   Never happens. (This is only the opinion of the speaker so it's not a fact) It's very unlikely to happen. It's going to take so long to actually happen. (It's so long that you'd be better off giving up expecting it)  3.1. We also use it when someone is always postponing something they have to do so you never actually see it happen.   For example:      Kyle: When are you going to pay my money back?      Cartman: When the  blooms.   Now I don't know if I got the plant's name right because there are some other versions like:        In any event, they have mentioned this plant in the idiom because apparently some versions of it actually never bloom. Others believe it's because farmers harvest them before they get the chance to bloom (e.g to make music instruments), so it never happens. I'd love to know the equivalent idioms or expressions for this.  PS: I only know "When pigs fly" but I'm not a fan of it because the word pig sounds a bit rude to me and partly because it implies a strong impossibility which doesn't work for the #3 use. Can I color text by using an image, so that the color isn't flat? I'm using tikz to attempt to make a book cover, and I have a "wallpaper" image that looks like leather (not that I've handled that many leather-bound books). I'm using the node command to place text over the top of it, but none of the colors I've experimented with look decent.  Can I use an image of gold-leaf as the fill/stroke for the text? Or for that matter, for the path lines as well?  I've tried googling for this, but I can't figure out a set of keywords non-generic enough to pull back useful results. Should spam posts be edited? Using Squeeze Theorem to show the following limits Consider a function f (x) defined on R satisfying:   for all x distinct 0.  Calculate: lim f(x) with x→∞ and limf(x) with x→−∞   Issue information:   if x → ∞ then we can consider | x | = x  if x → -∞ then we can consider | x | = -x  I can use x² = | x |? Is $\sum\limits_{n=1}^{\infty}\frac{n!}{n^n}$ convergent? I have been trying to figure out the time complexity of the conjugate gradient method.  I have to solve a system of linear equations given by  $$Ax=b$$  where $A$ is a sparse, symmetric, positive definite matrix.  What would be the time complexity of the conjugate gradient method? Disabling Intel Turbo Boost in ubuntu Do the topologies with the same convergent sequences coincide? Do the same sequential convergence in two topologies the result are the same topologies? Let $X$ be topological  space with topologies $\tau$, $\tau'$. Let $(x_{n})\in X$ and $x\in X$ and $(x_{n})\rightarrow x$ with $\tau$ if and only if $(x_{n})\rightarrow x$ with $\tau'$. Do the two topologies on $X$ coincide? How do I fill an area with faces lined up in a grid over a not flat surface? Hello to everyone again,  I'm looking for an addon or a short cut to fill fast and simmetrically a NOT FLAT surface that needs to be closed with vertices. Doing it manually it takes a lot of time. I tried to do CTRL F and then Grid fill,but it didn't work. Neither F key works. Check the attached screenshot. thanks. I am trying to install Ubuntu 16.04 LTS on a Dell XPS 13 (9350). The laptop came with Windows 10 pre-installed. I am installing from a pendrive but I think the issue would be the same if I used e.g. a USB DVD drive.   Dude, where's my hard drive?  I am blocked because the installation program cannot find 8.6Gb of free space on my hard drive despite the fact that there are hundreds of Gb free according to Windows. Digging a bit into the problem, it seems that Ubuntu cannot see my hard disk at all.    This is the output of GParted:    I am not sure what that drive mounted at /cdrom is, but it seems the only device that exists in the system that is not a RAM disk or a loop device:   Content of /proc/partitions:   ubuntu@ubuntu:~$ cat /proc/partitions   major minor  #blocks  name     1        0      65536 ram0     1        1      65536 ram1     1        2      65536 ram2     1        3      65536 ram3     1        4      65536 ram4     1        5      65536 ram5     1        6      65536 ram6     1        7      65536 ram7     1        8      65536 ram8     1        9      65536 ram9     1       10      65536 ram10     1       11      65536 ram11     1       12      65536 ram12     1       13      65536 ram13     1       14      65536 ram14     1       15      65536 ram15     7        0     130048 loop0     7        1    1104188 loop1     8        0    1970239 sda     8        1    1953439 sda1   Summary of fdisk -l:   ubuntu@ubuntu:~$ sudo fdisk -l |grep Disk Disk /dev/ram0: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram1: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram2: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram3: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram4: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram5: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram6: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram7: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram8: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram9: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram10: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram11: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram12: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram13: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram14: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/ram15: 64 MiB, 67108864 bytes, 131072 sectors Disk /dev/loop0: 127 MiB, 133169152 bytes, 260096 sectors Disk /dev/loop1: 1.1 GiB, 1130688512 bytes, 2208376 sectors Disk /dev/sda: 1.9 GiB, 2017525248 bytes, 3940479 sectors Disklabel type: dos Disk identifier: 0x00000000   Result of fdisk of the only non-loop non-RAM device (not sure this is relevant):   ubuntu@ubuntu:~$ sudo fdisk -l |grep -A 50 sda Disk /dev/sda: 1.9 GiB, 2017525248 bytes, 3940479 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 Disklabel type: dos Disk identifier: 0x00000000  Device     Boot Start     End Sectors  Size Id Type /dev/sda1  *      129 3907007 3906879  1.9G  c W95 FAT32 (LBA)   I am blocked at this point. Any help will be much appreciated.   Additional information (following comments):   Running dmesg |grep -i sata shows (among others) eight lines that suggest some kind of error:   ata1: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata2: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata3: SATA link down (SStatus FFFFFFFF SControl FFFFFFFF) ata4: SATA link down (SStatus 0 SControl 0) ata5: SATA link down (SStatus 0 SControl 0) ata6: SATA link down (SStatus 0 SControl 0) ata7: SATA link down (SStatus 0 SControl 0) ata8: SATA link down (SStatus 0 SControl 0)  I have "Restarted" windows instead of only "Shutdown" it to avoid problems with "fast startup", following some recommendations I have read of other users having problems installing Linux on Windows 10 laptops.ago   Regarding SATA, the BIOS says: "SATA Operation: RAID ON". The other two options available are "Disabled" and "AHCI".    I have followed the instructions of these two pages from Dell: , .