Rietz-Fischer Theorem: Let $E$ be a measurable set and $1 \le p \le \infty$.  Then every rapidly Cauchy sequence in $L^p(E)$ converges both with respect to the $p$-norm and pointwise almost everyone on $E$ to a function in $L^p(E)$.  Question: How does one prove this in the case that $p = \infty$?  Every proof I have seen thus far seems to assume that $p \in [1,\infty)$.  For example, the  makes this assumption and then concludes by remarking "the case $p = \infty$ reduces to a simple question about uniform convergence".  Upon thinking about this for a bit it's not obvious how to proceed.
I'm reading page number 4 here. In particular the section where it deals with the case $p=\infty$, that is , showing that $L^{\infty}$ is complete.    Two questions:  1) Why is the convergence uniform? where it says "for $x \in N^{c}$ , $f_{n}$ is a Cauchy sequence of complex numbers. Thus $f_{n} \rightarrow f$ uniformly. Clearly we have pointwise convergence but why is it uniform?  2) I don't see why $||f_{n} - f||_{\infty} \rightarrow 0$. Can you please explain this step?  Thanks.  (3/2015) Edit: The original link appears to be broken.  seems to provide a similar (maybe even identical) proof to the one the OP talks about with slight notational differences.
When you tap a row in a UITableView, the row is highlighted and selected. Is it possible to disable this so tapping a row does nothing? Does math have an incorrect notation / syntax? I don't mean writing misaligned notation (google), but when you take something like a number to powers to powers to powers, $${{2^2}^2}^3$$ (I was told this is incorrect notation by a teacher). Is it really incorrect, or does it just need to be simplified with parentheses? Do people write maths like this?  a radical expression with the root being a radical expression? $$\sqrt[\sqrt{2^3}]{2}$$ finding rational roots of polynomials Could someone please explain me how to apply the Descartes's Criterion? For example , how do I find the rational roots of $ x^3 -x +1$.  I've been looking at some examples, but I get confused. How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$? How many ordered pairs $(A,B)$ they are if $A$ and $B$ are subsets of $\{ 1,2,3,4,5 \}$ such that $A \cap B = \{1\}$?  If 1 is in both sets :  Size of $A$ is $&gt;1$ and size of $B$ is $&gt;1$.  $\Rightarrow$ Size of $A \cup B =4$  $$5 \cdot (3 \cdot 3 \cdot 3 \cdot 3) = 405$$ is not correct.  What am I missing here? How to include audio file in PDF How do I include audio file, or should I say, link to audio file in PDF?   Audio should be playable "from PDF directly", without opening any additional window. Is it possible to attach audios to PDF     then so I can manipulate only with     that file, without being bothered     with copying folders where those     audio files would reside?   I saw that 1. is definitely doable (but don't know how) and I'm interested if second one is also possible (while compiling .tex file) since I know that some files (like graphs and tables) can be attached in PDF, but after it's generated.   Button for playing audio from document that uses this trick. How to batch watermark JPG files? I have around 15,000 JPG files that need to be watermarked. Is there any way to process these images en masse and add a watermark?  The images may be different sizes, I want the watermark placed in the center of the images. How often did Gandalf return to Hobbiton after the events of The Hobbit? Feature selection with Random Forests How to stop mysql from running at boot time? Currently, my MySQL server starts on every server boot. For a couple reasons, this is undesirable behavior. Is there a way to disable this behavior? Do I say "I study computer science," or "I study Computer Science"? Similarly, "I really liked that computer science course," vs. "I really liked that Computer Science course." has anybody been successful getting pandas to work with QGIS's Python installation, on Windows? I've used the OSGeo4W installer to install QGIS2.2, which comes with Python 2.7 and Numpy 1.5.1. Any reasonably new version of pandas requires Numpy 1.6.1 or newer. So that seems like my main problem. I've tried: - Building an older version of pandas (e.g. 0.7.3 and 0.4.1) using easy_install, but that gives me a very long list of errors, at least the last of which are related to distutils\msvc9compiler.py - Using binaries of old pandas versions, but they are all built for Numpy 1.6.1 - Upgrading Numpy, which also gives me an error message. In case anybody is interested, the message ends with "File “C:\OSGeo4W\apps\Python27\lib\distutils\msvc9compiler.py”, line 299, in query_vcvarsall” newline “raise ValueERror(str(list(result.keys())))” newline “ValueError: [u’path’]"  I'm wondering if anybody else has ever been successful in getting pandas to work with a QGIS install on Windows??  It would be so easy if QGIS would ship with, say, a version of Numpy that's less than three years old. Does anybody know if such an upgrade is in the works for upcoming QGIS versions? Equivalent of *Nix 'which' command in PowerShell? How do I ask PowerShell where something is?    For instance, "which notepad" and it returns the directory where the notepad.exe is run from according to the current paths. I'm having some trouble understanding cosets.  If I understand it correctly, cosets form a partition. So, if I have $|G:H| = 2$, then $G = H \cup xH$. Right?  In an exercise, I'm asked the following:  "For $G$ group and $H \le G$ such that $|G:H| = 2$, prove that $x^2 \in H$ for any $x \in G$"  If $x \in H$, then $x^2 \in H$ because $H$ is a group. But, if $x \notin H$, then $x \in gH$ for any $g \in G$, right? Then $x^2$ = $(gh)^2$ but I can't see why this is in $H$.  Can you give me some hint? Thanks for stopping by. Sorry if this is a stupid question, but I am mainly working with 5V PIC circuits and now want to integrate a component (nRF24l01+) that needs a 3.3V supply.  As I have a box full of ~7V wall-warts, I normally use a 7805 or similar LDO regulator to power the circuit. Now I want to also get 3.3V without using a separate external supply.  I'm looking for guidance on the "right" way to do this. I think the options are:   Use a voltage divider on the output of my 5V regulator to get 3.3V Use a 3.3V LDO regulator fed directly from the 7V wall-wart   Use the output of the 5V regulator as input to a 3.3V regulator   I can see that all of these would work (would they?) and that option 3 seems to waste least power, but I don't know what gotchas might arise by chaining two regulators like this.  What things do I need to consider when making this choice? I am flying from Orange County, California to San Francisco, California and then on to Seoul, South Korea. If I check a bag will I have to pick it up in San Francisco and re-check it and go through security or customs for the international flight?  Everything I am reading leads me to believe that it's only necessary to re-check baggage and go through security and/or customs when you are first arriving into a country. However, since my first leg is staying within the United States I assume the airline will transfer my bag to the next flight and I can just go straight to the gate for my international flight?  Both legs of the flight are through the same airline (United Airlines) and there are only the two legs: OC -> SF, then SF -> Seoul. How are cellphone IP addresses assigned? Downcasting in Java Upcasting is allowed in Java, however downcasting gives a compile error.   The compile error can be removed by adding a cast but would anyway break at the runtime.   In this case why Java allows downcasting if it cannot be executed at the runtime? Is there any practical use for this concept?   public class demo {   public static void main(String a[]) {       B b = (B) new A(); // compiles with the cast,                           // but runtime exception - java.lang.ClassCastException   } }  class A {   public void draw() {     System.out.println("1");   }    public void draw1() {     System.out.println("2");   } }  class B extends A {   public void draw() {     System.out.println("3");   }   public void draw2() {     System.out.println("4");   } } Deriving curvature formula How do you derive the formula for unsigned curvature of a curve $\gamma (t) = (x(t), y(t)$ which is not necessarily parameterised by arc-length.   The formula given is $$|\kappa (t)| = \left|\frac{x'' y' - y'' x'}{((x')^2 + (y')^2)^{3/2}}\right|$$  All the definitions of curvature in my notes give it in terms for an arc-length parameterisation $T'(s) = \kappa(s) N(s)$. This is apparently meant to be 'easily derivable', am I missing something? Preventing itemize environment to insert initial vertical space I'm using itemize and tables like the following  \begin{tabularx}{\textwidth}{&gt;{\ttfamily}llX}  \toprule  \textbf{\rmfamily Operator} &amp; \textbf{Typen} &amp; \textbf{Regel} \\  \midrule  = &amp; Alle Typen &amp;   \begin{itemize}    \item This is how we roll    \item Maybe we can talk?   \end{itemize} \\  \bottomrule \end{tabularx}   But it keeps inserting a vertical space before the itemized list. How can I prevent it from doing so? How well does T-Mobile's (US) Simple Choice plan work when in Europe?