Cauchy Schwarz inequality for $\langle f,g\rangle = \int_0^1f(x)g(x)\mathrm{d}x$ For the vector space of continuous functions on $[0,1]$ Define the inner product as  $$\langle f,g\rangle = \int_0^1f(x)g(x)\mathrm{d}x$$ Please help me to prove the Cauchy Schwarz inequality for this given inner product.  Cauchy Schwarz Inequality: $|\langle v,u\rangle|\leq \lVert v\rVert\lVert u\rVert$ for the elements $v,u$ in the inner product space.
Cauchy-Schwarz inequality proof (but not the usual one) Before you downvote/vote-to-close, I am not asking for a proof of: $$\sum^n_{i=1}a_ib_i\le\sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum^n_{i=1}b_i^2 } $$ Which is what EVERY link I've found assumes is the inequality (for a proof of that see:  )    I am reading a book that claims the Cauchy-Schwarz inequality is actually: $$\vert\langle x,y\rangle\vert\le\Vert x\Vert\Vert y\Vert$$ where $\Vert x\Vert :=\sqrt{\langle x,x\rangle}$  with the additional claim: equality holds $\iff\ x,y$ are linearly dependent  I cannot find a proof of this claim (only proofs for the dot product inner product). My question is: what is the simplest way to prove this (I define simplest to mean "using as few definitions outside of the inner product itself as possible" - so that's not including Gramm-Schmitt orthonormalisation, for example.)  I've just found some possible answers in the links on the right (annoyingly) but fortunately I have a second, albeit softer, question:  What inequalities that are common for say $\mathbb{R}^n$ are actually based on the inner product?   (In future I will check this site first, I was targeting lecture notes and such in my search)  Addendum: I want to use this on not-finite dimensional vector spaces. I've found one proof that relies on finite-ness.
I'm not sure if this is the right place to ask this question. I realise that this maybe a borderline philosophical question at this point in time, therefore feel free to close this question if you think that this is a duplicate or inappropriate for this forum. Anyway, I'm an electrical engineer and I have some basic knowledge of quantum mechanics. I know that Schrödinger's equation is deterministic. However, quantum mechanics is much deeper than that and I would like to learn more. If this question is not clearly answerable at this point than can anyone point out some recognized sources that try to answer this question. I would appreciate it if the source is scientific and more specifically, is related to quantum theory. How do I play a YouTube video in the background on my iPad? Does a Warlock receive the benefit of their familiar's Magic Resistance trait? The Warlock has access to special forms for their find familiar spell through the Pact of the Chain feature. I'm wondering if these special forms, the Quasit, Imp, and Pseudodragon in particular, allow the player character to share the familiar's Magic Resistance feature. In the Monster Manual each of these creatures has a sidebar that states that the familiar shares its Magic Resistance feature with the companion they are bonded to, but the PHB doesn't mention this in any of the creatures' stat blocks.   For example, in the Variant: Pseudodragon Familiar sidebar on page 254 of the MM it says:     "While the pseudodragon is within 10 feet of its companion,        the companion shares the pseudodragon's Magic Resistance trait."   This feature seems clear in the MM but I'm led to believe that it wasn't intended for Player Characters since mention of it is absent from the stat blocks in Appendix D of the PHB (pages 307-309).  Would a 3rd level warlock who chose one of these familiar forms through the Pact of the Chain feature benefit from advantage from saving throws from spells and other magical effects due to these special forms, or was this feature only meant for powerful NPCs and enemy spellcasters who had made this link with these familiars themselves? What is a tty, and how do I access a tty? I'm having issues with my macbook pro OS X 10.7.5.  It used to connect to the wifi router (D-Link N300, DIR=615) no problems.  But for the past several weeks, I haven't been able to at all.  What pops up when I select my wifi network from the list:   "Failed to join "network" A connection timeout occured".   What happens when I try to enter my password manually:   "Connection timeout"   What I've tried to do as solutions:   Google Restart router by pressing reset &amp; refresh button Delete System &amp; User (Login) Keychain access Network > Gear symbol by the + and - bottom left of the screen > Set Service Order > Drag Wifi to the top Session ID from scheduled jobs Here's the scenario.  I have a remote application which is invoked from an HTTP callout.  In this callout, I am passing the session ID.  The application can then do its stuff without needing a username and password.  However, I want to be able to invoke the application from a scheduled job, and getsessionid() returns null in this case.  Is there any way around this?  I am trying very hard to avoid storing sfdc usernames and passwords on my remote server, and I want to avoid the hassle of using oAuth with all the complications of client secrets, etc. What is the speed of sound in space? Given that space is not a perfect vacuum, what is the speed of sound therein? Google was not very helpful in this regard, as the only answer I found was $300\,{\rm km}\,{\rm s}^{-1}$, from , which is not a source I'd be willing to cite. What is meant by "obfuscate by pretending"? I came accross the following sentence:  "Politicians obfuscate by pretending that the world would be the same if it weren't for the Cold War."  What is the meaning of obfuscate by pretending and how does it relate to the rest of the sentence?  The way I understood it: politicans are aware that the world would not be the same if it weren't for the Cold War, but are (due to unknown or suspected reasons) not willing to admit this and therefore pretend otherwise. Reason for Subject-Verb Inversion: Only in cases where A is B, shall the Company do X Should we have a tag for significant points in time? There are many discussions about the current incidence. For example:          Should we have a whole tag for this incidence? In general, if in the future there are more crises and incidences like this, should they have their own tags too? I feel it's fine, as we have specific tags for individual events like . And, should we have a  tag?    Related:        Note that these questions are specifically about the conflict 2019, while this question is simply about creating meta tags for significant events. Any thoughts on a semidecadian badge for people with five years service and an average of N points per year? Given that Stack Overflow is close to its fifth birthday as a public site, has any thought been given to a 'semidecadian' badge for people who have:   been registered with the site five years (a semi-decade, hence the name), scored more than N points per year on average (where N might be 1000).   The idea behind 'average of N points per year' is to allow slow starters to gain the badge eventually, rather than ruling them out in year 1.  The details are obviously subject to tweaking.  This might be a gold badge (the platinum version would be the decadian badge, with a larger value of N required — and no-one eligible for another 5 years or so).  These would complement the current Yearling badge (which is silver). How to reduce the spacing between responses in the alterqcm package? This question has a continuation here   The alterqcm package is no longer maintained by Alain Matthes. Two other developers maintain it. I wrote to them 15 days ago by email to report the error reported here:  I have not received a reply. I am trying to reduce the spacing between the lines of the QCM as indicated in  by @Werner and it does not work. In addition, by using the David Carlisle's code given in , I get an X instead of a square.  Full code of the screenshot: \documentclass[12pt]{article} %\usepackage[utf8]{inputenc} %\usepackage[upright]{fourier} \usepackage{etoolbox}% http://ctan.org/pkg/etoolbox \usepackage[english]{alterqcm} \def\square{X} % &lt;---------- David Carlisle comment \def\nogreekalph{??} % &lt;---- David Carlisle comment \parindent0pt \begin{document} % Original display \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \bigskip  % Smashed display !!!!!! Dont work now  {\begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {\smash{Proposition 1}},  {\smash{Proposition 2}},  {\smash{Proposition 3}}} \end{alterqcm}}  \bigskip  % Global smashed display !!!! Dont work now  \makeatletter \patchcmd{\aq@prop}{\vbox{#1}}{\vbox{\smash{#1}}}{}{}% patch \aq@prop \makeatother \begin{alterqcm}[lq=8cm]  \AQquestion{Question}{%  {Proposition 1},  {Proposition 2},  {Proposition 3}} \end{alterqcm}  \end{document}  Is there a solution? Consider the sum of $n$ uniform distributions on $[0,1]$, or $Z_n$. Why does the cusp in the PDF of $Z_n$ disappear for $n \geq 3$? I've been wondering about this one for a while; I find it a little weird how abruptly it happens. Basically, why do we need just three uniforms for $Z_n$ to smooth out like it does? And why does the smoothing-out happen so relatively quickly?  $Z_2$:     $Z_3$:    (images shamelessly stolen from John D. Cook's blog: )  Why doesn't it take, say, four uniforms? Or five? Or...? Can a wizard using an arcane focus quarterstaff and a shield still cast somatic component spells? If there is a wizard that, by some means, has acquired proficiency in the use of shields, can said wizard use his shield in one hand and his arcane focus quarterstaff in the other and still cast somatic component spells?  Since the quarterstaff has been crafted to also be an arcane focus, and the Sage Advice Compendium states that you can provide the somatic components of spells with the same hand wielding the arcane focus, in this case an arcane focus quarterstaff, could that wizard still be able to perform somatic component spells while using a shield in the other?  I understand War Caster is the feat that allows you to cast somatic component spells even while using a shield and weapon or two weapons in each hand, but is this a simple way around those rules seeing as this "weapon" is your arcane focus? I would like to believe it is not, but I'm not fully fluent in the ways of spellcasting. What characters are valid in a Java class name? What other rules govern Java class names (for instance, Java class names cannot begin with a number)? How to implement a \tomorrow in latex I was asked how to implement a \tomorrow macro in TeX/LaTeX. While I could come up with a reasonably good solution, working for one document class, I don't see how that could be achieved in a document class agnostic way.  I found , but even though the date arithmetic may be of use, it assumes a specific format for the date (which also is wrong for the locale I reside in).  Does anyone have a reasonably portable solution? Is this even possible? I am looking for an Ubuntu version that consumes less Memory and CPU. I have read both Lubuntu and Xubuntu (The homepages, wikipedia, phoronix and other sites comparing both). But from experience, which one uses less memory and is less CPU intensive. I need to install them in very old hardware and want to persuade the owner of the hardware of the benefits of Ubuntu.  in this case I want to install 11.10 or 12.04 when it comes out. How are each behaving in those versions?  The 2 PCs I will be installing either Xubuntu or Lubuntu are:  Granpa PC:  CPU - Pentium 2 450Mhz RAM - 64MB DIMM Video - 16MB Used for - Documents and Internet. No listening to music, no looking at videos. Just using it for document writing.  The other old meat:  CPU - Pentium 3 550Mhz RAM - 128MB DIMM Video - 16MB Used for - Documents and Internet also but they want.. or maybe they are wishing for it to use it to see movies and listen to music. This one has internet. The other one does not.  UPDATE  Wanted to update this question with the test results for over 8 months worth of using Lubuntu and Xubuntu. When it comes to very old hardware, am talking about 64 MB, 128 MB or even 256 MB of RAM computers, Lubuntu is better than Xubuntu when comparing Memory usage. When it comes to CPU usage, both are the same.  This will not improve I/O Disk performance which will be the bottleneck for old computers. Also the difference in performance between Xubuntu and Lubuntu is about 10% to 20% percent in memory usage. Again, CPU usage is the same.  Now, when it comes to how responsive the Desktop is, LXDE (Lubuntu) has a much better response time than XFCE (Xubuntu). I would say about 50% or just simply noticeable to the eye. Lubuntu is similar to how Windows XP looks while Xubuntu is more like how Gnome 2 looked.  The computers (The old ones mentioned here) and 2 more are still working in excellent conditions thanks to the Ubuntu family. I also want to add I am using 13.10 for both distros. How do I calculate the date six months from the current date using the datetime Python module? I am using the datetime Python module.  I am looking to calculate the date 6 months from the current date. Could someone give me a little help doing this?  The reason I want to generate a date 6 months from the current date is to produce a review date.  If the user enters data into the system it will have a review date of 6 months from the date they entered the data. Let $Z_n$ be the set of all $n \times n$ matrices that commute with all $n \times n $ matrices. Show that $$Z_n = \{\lambda I_n \ | \  \lambda \in \mathbb R\}$$  ($I_n$ is the $n \times n$ identity matrix)  I don't know how to use $E_{ij}$ (matrix with $1$ in $(i,j)$ and $0$ elsewhere) and the elementary matrix $P_{ij}$ to prove this question. Can anyone explain it please? Command line option to check which filesystem I am using? Is there a command that will show which file system (ext3, ext4, FAT32, ...) the various partitions and disks are using?   Similar to how sudo fdisk -l lists information about disks and partitions?