Let $(\Omega,\mathcal A,\mu)$ be a measure space, $p,q\ge1$ with $p^{-1}+q^{-1}=1$ and $f:\Omega\to\mathbb R$ be $\mathcal A$-measurable with $$\int|fg|\:{\rm d}\mu&lt;\infty\;\;\;\text{for all }g\in L^q(\mu)\tag1.$$ By $(1)$, $$L^q(\mu)\ni g\mapsto fg\tag2$$ is a bounded linear fuctional and hence there is a unique $\tilde f\in L^p(\mu)$ with $$(f-\tilde f)g=0\;\;\;\text{for all }g\in L^q(\mu)\tag3.$$     Can we conclude that $f=\tilde f$?   EDIT: As we can see from , we need to impose further assumptions; but which do we really need?
Let $f$ be a measurable function on a measure space $X$ and suppose that $fg \in L^1$ for all $g\in L^q$. Must $f$ be in $L^p$, for $p$ the conjugate of $q$? If we assume that $\|fg\|_1 \leq C\|g\|_q$ for some constant $C$, this follows from the Riesz Representation theorem. But what if we aren't given that such a $C$ exists?
Every time Windows 10 runs updates (which I see no way to control), it updates my graphics driver.  I like the old one because it is faster.  So, I have to reinstall it.  This is getting to be a pain.  Is there any way to stop Windows 10 from updating it again? Verbless sentence Why does $\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$? Apery's original proof that $$ \zeta(3) \equiv \sum_1^\infty \frac1{n^3} $$ is irrational starts from an alternating series  $$ \zeta(3) = \frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}} $$  There must be a way to see that those two series are equal, but I have tried various telescoping and other techniques and I just can't see it. Help me out:       Show (preferably by reasonably elementary means) that$$\sum_1^\infty \frac1{n^3}=\frac52\sum_1^\infty\frac{(-1)^{n-1}}{n^3\binom{2n}{n}}$$ about how to open a URL in your default browser using the Terminal. Basically, you use the command   xdg-open http://google.com   My default browser is Firefox. But lets say that I wanted to open a specific website in Chrome, instead of Firefox. Is there a command to open it in Chrome instead of Firefox? Why is posterior density proportional to prior density times likelihood function? According to Bayes' theorem, $P(y|\theta)P(\theta) = P(\theta|y)P(y)$. But according to my econometric text, it says that $P(\theta|y) \propto P(y|\theta)P(\theta)$. Why is it like this? I don't get why $P(y)$ is ignored. Logistic regression cost surface not convex I am building a simple logistic regression model on 2D data. Here is the input I use.    I built a logistic regression model using this data and it successfully is able to find the discriminating line between these two classes. Here is the separating line found by my program:    Also, just for debugging, I plotted number of iterations vs J(theta), which also looks fine:    Now, confusing part comes into play. I am investigating the cost surface of this model that the logistic regressions is doing the gradient descent on. I am considering three different loss functions. Sum-of-squared loss, hinge-loss, log-loss.  Hinge loss:    Log-loss:    Sum-of-squared loss:    As far as I know, both hinge loss, sum-of-squared loss and log-loss should produce a convex surface. When I look at my surfaces, however, sum-of-squared surface does not look convex to me. Why is that the case?  For convenience, here is the two critical functions from my code:  function retval = J(X, y, theta)     M = size(y, 1);     % sum of squared error     retval = (1/(2*M)) * sum((h(X, theta) - y).^2);  endfunction  function retval = h (X, theta)     % logistic     retval =  1 ./ (1+ exp(-1 * (X * theta))); endfunction Why is it a bad practice to lock the object we are going to change? Including Bitmap (.bmp) I can't include bitmaps with pdflatex. Here is my mwe:  \documentclass[12pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[T1]{fontenc} \usepackage[turkish]{babel} \addto\extrasturkish{\uccode`i=\string"9D \lccode`I=\string"19 } \addto\noextrasturkish{\uccode`i=`I \lccode`I=`i }   \usepackage{graphicx}  \begin{document}  \shorthandoff{=}  \includegraphics[width=5cm,height=5cm]{000.bmp} \end{document}   Gives the error Unknown graphics extension: .bmp. Prevent pdflatex from writing a bunch of files If I create just a basic hello.tex file an compile it using   pdflatex hello.tex   The pdflatex program is creating a plethora of file which is a little annoying. Currently it writes all those files about which I really don't care:  hello.aux, hello.log, hello.nav, hello.out, hello.snm, hello.toc   Is there a way to tell pdflatex to only write hello.pdf if the build is successful ? If not, can I specify the output directory of those files ? Maybe I could just write them in /tmp. I am trying to create the following trigger:      CREATE TRIGGER update_customers AFTER INSERT ON wp_usermeta     FOR EACH ROW      BEGIN             if new.wp_metakey = first_name then insert into test_customer(first_name) values (new.wp_meta_value);     END   I keep getting a 1064 error that says, "You have an error in your SQL syntax; check the manual that corresopons to your MySQL server version for the right syntaxt to use near " at line 5"  What am I doing wrong? Traveling to the US with new passport, visa on the old one Which IP address do I use to access my server? I'm running a Samba server on my MacBook and I've also set up port forwarding to said server from my router.  My public IP is 106.51.x.x (according to google and a few other services) and my Mac's local IP is 192.168.x.x, but when I head to my router's configuration page, it says my external IP is 10.242.x.x.  This is what I don't understand, when connected to my home WiFi, using the 10.242.x.x address works just fine. But if I try to access the same over 4G, the connection times out.  As for the public address, neither WiFi nor 4G works.  Is there some setting on my Mac (I'm running MacOS Sierra) that I must alter to make it visible to the public? How do I get fan control working? I know there something called fancontrol, that enables you to control the speed of your system's ventilation. I'd like to let my fans spin a bit faster as my laptop is heating up very easilly. All tutorials and stuff I've found are for old versions of Ubuntu and don't seem to be working anymore.  Can anyone explain to me or give me a good link on how I  can get it working on Ubuntu? Something different with the same effect is also fine. How many switches can be stacked/interconnected? How do I enumerate the properties of a JavaScript object? Our group has determined that buying (almost exclusively) money and points is the way to go.  Most kingdom cards are ignored, or are bought very sparingly. This has lead to games feeling 'samey' and has taken a lot of the fun out for us.  What strategies should we use to counter ""?  Or are house rules needed here? Add layer to a QGIS group using Python I am trying to add new layers into a specific group which I can get to work if I know the index, but how can I get Python to find the index of a group so it can load the new layer into the group wherever it is in the TOC?  qgis.utils.iface.legendInterface().moveLayer( layer, index ) $\lim_{h\to 0}\int \frac{f(x+h)-f(x)}{h}\,\mathrm{d}x = f(b)-f(a)$ Windows Managers vs Login Managers Vs Display Managers Vs Desktop Environment I posted a  and noticed people weren't distinguishing correctly between many of these things: Windows Managers vs Login Managers Vs Display Managers Vs Desktop Environment.  Can someone please clear this up, i.e. tell us the difference between them and how they are related perhaps?  What category does Xorg fall under?  What about Gdm/Kdm/Xdm?  People also talk about X.  What is X? How Do I Compute the Eigenvalues of a Small Matrix? If I have a $2\times 2$ or $3\times 3$ matrix, how should I go about computing the eigenvalues and eigenvectors of the matrix?  NB: I am making this question to provide a unified answer to questions about eigenvalues of small matrices so that all of the specific examples that come up can be marked as duplicates of this post. See .