The norm of an element $(\gamma_k)=x \in l^{\infty}$ is defined to be $\sup_k \gamma_k &lt;\infty$. The norm of an element $(\gamma_k)=x \in l^p$ is defined to be $(\sum_k \gamma_k^p)^{1/p} &lt; \infty$, even for $p=1$!   I can't see how $\displaystyle{\lim_{p \to \infty}} (\sum_k \gamma_k^p)^{1/p} = \sup_k \gamma_k$ holds?    And, if they just are separate definitions, then why the case '$p=\infty$' is in the 'category' of $L^p$ spaces?   Added - Understanding answers to  requires a knowledge more than an undergraduate Analysis, i.e. my current level. Simple detailed answers would be much appreciated.
Could someone help me prove that given a finite measure space $(X, \mathcal{M}, \sigma)$ and a measurable function $f:X\to\mathbb{R}$ in $L^\infty$ and some $L^q$, $\displaystyle\lim_{p\to\infty}\|f\|_p=\|f\|_\infty$?    I don't know where to start.
Showing order of $Z (G) =1$ or $pq$ Can I update an Mac App purchased by a different Apple ID? What is the ideal way to handle data pins D+ and D- on a USB power adapter to be compatible with fast charging on devices? Does Stack Exchange contain any edit suggestion or bug submission system, or such type of discussion forum to inform SO or its administrators about suggestions and bug reports that relate to SO itself? Proof that the irrational numbers are uncountable Are The Lazarus Effect and Dune in the same universe? There are similar aspects between Dune, and the technology in The Lazarus Effect.   Noticeable similarities:   Lasguns - this weapon exists in both of the books, and has similar architecture in both books.   Merman and Ixians - are both technologically sophisticated in similar ways. Rely on mechanical technology as opposed to organics or similar.   Is this merely Herbert sticking to his original style? Is there a difference between single and double quotes in Java? Is it possible to obtain AES-128 key from a known ciphertext-plaintext pair? I have a file, which was encrypted with AES-128 in ECB mode. I know the format of the original file and know that all files in this format have the same headers. So, I have an encrypted block and the original block.  Can I get the encryption key, using this information? A (highfalutin) word for a highfalutin word? To illustrate the question, badroit used the word "meretricious" meretriciously.   Is there an English word that describes highfalutin words like "meretricious" that are (gratuitously) used in place of better-understood words like "garish"?  The word may or may not be self-describing. Auto mount a disk drive in ubuntu 14.04 LTS (Bookmark directory not found) I am trying to pickle some mangoes the Indian way. I boiled the jar in hot water, dried it and then added the mixture of spices, salt and mangoes to the jars after the mixture had cooled. I made the mixture in a frying pan to try and remove as much water as I could. After it was cooled, I put it in the jar and heated some mustard oil till it just barely started smoking. I let that cool and poured it over the mango spice mixture.  For the first three days it was fine. On the fourth day, I opened it to test the taste and ended up adding more salt. Now, there seems to be gas forming in the jar. I've noticed it a few times now. When I open the jar, I hear a whooshing noise as air escapes. It's quite a lot, since I opened the container about 3 times in a 10 hour period and each time gas escaped. Has this pickle gone bad? Or is this something that's normal? How do I convert materials from Blender Internal to Cycles? I know this may be a tough or confusing question, but I have numerous existing materials I created and reuse for a project, which were all prepared for Blender Internal.  Now, I want to start the transition to Cycles, but I've hit a roadblock. How can I transition my materials without completely starting over? Or should I just forget it and start with fresh materials? Suppose that I have a group $G$ of order $p^{2}q$ for two distinct primes $p$ and $q$.  I need to first show that one of its Sylow subgroups is normal.  I start by letting $H$ be a Sylow $p$-subgroup and $K$ be a Sylow $q$-subgroup.  If $K$ is not normal, then letting $r$ denote the number of Sylow $q$-subgroups, I have that $r|p^{2}$ and $r\equiv 1$(mod $q$).  I also have by the second Sylow theorem that $r\neq 1$.  So $r = p$ or $p^{2}$.  If $r = p^{2}$, I can write out the Sylow $q$-subgroups, and count their combined elements to show that $H$ is uniquely determined and then again imply Sylow number $2$ to obtain that $H$ is normal.  But if $r = p$, then I cannot see how to proceed.  Can anyone give any advice?  Thank you.  ( I am happy to provide more detail for the $r = p^{2}$ case if it is desired or appropriate. ) $x,y,z \in \mathbb R$ My attempt: Notice that if $x^2 + y^2 + z^2 \geq xyz$ and $x+y+z \geq xyz$. So : $$x^2 + y^2 + z^2 \geq x+y+z $$ But this is actually not always true, take the case when $ 0&lt;x,y,z &lt;1$ And you will see that this is not working. Maybe I’m wrong. How to construct an example of a probability distribution for which $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}$ holds, assuming $\mathbb{P}(X\ne0)=1$?    The inequality which follows from Jensen's inequality for a positive-valued RV $X$ is like $\mathbb{E}\left(\frac{1}{X}\right)\ge\frac{1}{\mathbb{E}(X)}$ (the reverse inequality if $X&lt;0$). This is because the mapping $x\mapsto\frac{1}{x}$ is convex for $x&gt;0$ and concave for $x&lt;0$. Following the equality condition in Jensen's inequality, I guess the distribution has to be degenerate for the required equality to hold. A trivial case where the equality holds is of course if $X=1$ a.e. Here is an example I found in a problem book: Consider a discrete random variable $X$ such that $\mathbb{P}(X=-1)=\frac{1}{9}, \mathbb{P}(X=\frac{1}{2})=\mathbb{P}(X=2)=\frac{4}{9}$. It is then easily verified that $\mathbb{E}\left(\frac{1}{X}\right)=\frac{1}{\mathbb{E}(X)}=1$.   This example shows that $X$ need not be positive (or negative) a.e. for the equality in the title to hold. The distribution here is not degenerate either.  How do I construct an example, possibly like the one I found in the book? Is there any motivation? Retrieve the position (X,Y) of an HTML element relative to the browser window I want to know how to get the X and Y position of HTML elements such as img and div in JavaScript relative to the browser window. In the  there are lots of numbers on display, including 1000, 2500 or 10000. The display is erratic though:    Same number, two displays. Both appear valid to me: 1000 or 1,000 but please, make it consistent. My eyes hurt! :) I just found , but there it's discussing the whole network while I'm asking just for that specific page. For a start. Create a bootable USB on a Mac to install Ubuntu on a PC USB-C Dongles killing Wifi on 2016/2017 MacBooks – Who’s to blame? I recently switched from my beloved MacBook Pro Retina (Late 2013) i5 2.6/16/512 to MacBook Pro (2017) Non-TB i5 2.3/16/512.  I purchased what from the build-quality looked like a good USB-C-to-DisplayPort cable with 4k/60Hz support, just to run into the very problem the Logicboard repair guru Louis Rossmann had described in : As soon as you plug in the USB-C dongle, the (2.4 GHz) Wifi stops working despite full antenna signal. Unplug the USB-C connector and you are immediately back in the game.  The big question I'm asking myself now: Who is to blame? Is this a design flaw in the current generation MacBooks with improper shielding or is it a design flaw in the various dongles?  I mean the fact that even Apple’s very own USB-C Digital AV Multiport Adapter is causing these WiFi issues (as you can learn from the many horrific ; but AFAIK have been fixed through a ), show that there’s something going horribly wrong. Wrong numeration in toc longer then one page