$f_n, f \in L^1$ and pointwise convergence implies ... Let $f_n, f \in L^1(\mathbb{R}), f_n \to f$ pointwise. Then $\int \left|f_n-f\right| \to 0$ iff $\int \left|f_n\right| \to \int \left|f\right|$.    My attempt was to use the Triangle Inequality for one side. Certainly taking convergence in mean as true we have convergence in mean of the absolute values, because of the triangle inequality.   Now, take convergence in mean of the absolute values. How to show convergence in mean of the original functions? I'm not even sure where pointwise convergence comes into play at this point. The only thing I tried was Egoroff's Theorem, and that led to nothing.   I would appreciate a hint, or a link if this is a repeated problem (I have a feeling it's standard, but I couldn't find it in the drop-down menu when I typed out the title).
If $f_k \to f$ a.e. and the $L^p$ norms converge, then $f_k \to f$ in $L^p$ Let $1\leq p &lt; \infty$. Suppose that    $\{f_k\} \subset L^p$ (the domain here does not necessarily have to be finite),  $f_k \to f$ almost everywhere, and  $\|f_k\|_{L^p} \to \|f\|_{L^p}$.    Why is it the case that $$\|f_k - f\|_{L^p} \to 0?$$   A statement in the other direction (i.e. $\|f_k - f\|_{L^p} \to 0 \Rightarrow \|f_k\|_{L^p} \to \|f\|_{L^p}$ ) follows pretty easily and is the one that I've seen most of the time. I'm not how to show the result above though.
Is there a constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$? Assuming the Axiom of Choice, every vector space has a basis, though it can be troublesome to show one explicitly. Is there any constructive way to exhibit a basis for $\mathbb{R}^\mathbb{N}$, the vector space of real sequences? Alignment issue in 'Your Sites' section in the Android app What are some examples of awkward sounding but grammatically correct sentences? What are some examples of awkward sounding but grammatically correct sentences? According to , the SE chat sites may have individual Easter eggs, depending on the site. If you find them, please post. Because it's so easy to make this stuff up, a screenshot as proof would be nice. (Although Photoshop isn't that hard either...) In plain English, how does casting spells and using creature abilities work with the stack and priority? I want to create a DEB file manually. I would like to just provide a folder which contains data to install, and a script to be executed after installation.  Is this possible? I have a Samsung Galaxy S2, running the latest CM10.1.  I lost the app drawer icon on my Dock; I must have deleted it. The app drawer icon lets you go to the app list, and the widget list.  How do I get the app drawer icon back? The icon looks like a circle with 6 dots inside it. /proc/PID/fd/X link number Why is the upper Riemann integral the infimum of all upper sums? I was reading the theory of Riemann integration when I cam across the following ,   If $f$ is bounded on $[a,b]$, and $P = \{x_0,x_1,x_2.......x_n\}$ is a partition of $[a,b]$, let $$M_j = \sup_{x_{j-1}\leq x\leq x_j}f(x)$$ The upper sum of f over P is $$S(P) = \sum_{j=1}^{n} M_j(x_j-x_{j-1})$$ and the upper integral of $f$ over $[a,b]$, denoted by $$\int_{a}^{b^-} f(x)dx$$ is the infimum of all upper sums.  The theorem similarly goes on to state the result for lower sums.  My doubt is : I do not understand how is $$\int_{a}^{b^-} f(x)dx$$ the infimum of all upper sums. I understand that if we refine the partition P, then the upper sum would decrease, so it may be a lower limit for all the upper sums computed on the refinements of P (but still being the lower limit does not prove that it is the infimum) and what about those partitions for which P itself is the refinement of?  How do I know that it will be a lower limit for those, let alone a infimum? I really like the new sidebar. With the new top bar eliminating the &quot;Hot Questions&quot; area there, I think this is the perfect place for it. One-click access to time-wasting questions about topics I may or may not have an interest in. Seriously, it's a procrastinator's dream come true. The only suggestion I can add is to extend it. I assume the &quot;hot&quot; algorithm can spit out more than 12 questions, and since we now have a huge amount of room for it(without being restricted to a weird drop-down scrollbox), why not show more? If there's something else planned for that spot, disregard. At a minimum, the title could link to the , as ShaWizDowArd suggested in comments. I've been using SE for a couple years now, and I've never seen that page, so it would add visibility there.    Update: They added a link! Just noticed this, I'm pretty sure it wasn't there yesterday. When you click it, it loads 12 more. At the moment, the link disappears once clicked, so you can see a total of 24 here now. Not a link to the full list, but better: why can't I edit /proc/sys/kernel/perf_event_max_sample_rate with vi I'm root, running Centos8 on bare mental and permissions on /proc/sys/kernel/perf_event_max_sample_rate are as follows  -rw-r--r--. 1 root root 0 Oct  3 14:00 perf_event_max_sample_rate   I want to change the sample rate by opening this file with vi, making the change and then writing it with either w or w!. When I do so it comes back with  perf_event_max_sample_rate" E667: Fsync failed  When I try to quit, it tells me  E37: No write since last change (add ! to override)   So I can't change the value with vi.  However I can change the value with   echo 4000 &gt; /proc/sys/kernel/perf_event_max_sample_rate   Is this simply because /proc is not a real file system and that all changes to things in /proc would have to be made using echo with a redirect, or maybe sed?  Is this behavior documented somewhere? Best compiler warning level for C/C++ compilers? Which is Most TimeMachine-like Backup Program for Ubuntu? I know this  but there is no comparison to OSX's TimeMachine.  I would like to find a backup software which is about the same as TimeMachine or better.  I would like to have such a software which can use some format of HDD which is suitable for big files (50 GB) in Linux.  I noticed that   OSX format of HFS+ is not supported by Linux distros, since no maintainer of HFS in Linux since 2006 NTFS format made by Debian 8.1 is not supported by older Linux distros   If Windows support, good.   Try with the Backups (Deja Dup) software  I did start the backup first time but I wanted to go to lunch after backup was running already two hours of just 10 GB files.  So it gave me this after the lunch    which is really insane, reported now , since the software is not designed to have pauses in doing the backups. Aborting a shell script if any command returns a non-zero value Why doesn't a bike/bicycle fall if going with a high speed? Is there any way to use a nether portal to travel to y = 250? magit: how to diff the same file between two different commits on the same branch? What do viruses do? There are those big green spiked things that my friends call viruses. I don't know what they do and I want to know. Do the viruses do something to the device you are playing on? Are they good or bad? Are hand motions required to use the Force? In almost every canon (that is, visual) source, Force users typically wave a hand to invoke the Force to move or manipulate objects, people and thoughts. I get out-of-universe this is a visual cue that the Force is being used, but in-universe is it strictly necessary?  (The only exception I can think of is when Luke is training on Dagobah and is balancing rocks while standing on one hand with Yoda on his foot. He doesn't appear to be waving his hand to move the rocks.)  Is this addressed anywhere in-universe, even in Legends? Ball to Ball Collision - Detection and Handling With the help of the Stack Overflow community I've written a pretty basic-but fun physics simulator.    You click and drag the mouse to launch a ball.  It will bounce around and eventually stop on the "floor".  My next big feature I want to add in is ball to ball collision.  The ball's movement is broken up into a x and y speed vector.  I have gravity (small reduction of the y vector each step), I have friction (small reduction of both vectors each collision with a wall). The balls honestly move around in a surprisingly realistic way.  I guess my question has two parts:   What is the best method to detect ball to ball collision? Do I just have an O(n^2) loop that iterates over each ball and checks every other ball to see if it's radius overlaps? What equations do I use to handle the ball to ball collisions? Physics 101 How does it effect the two balls speed x/y vectors?  What is the resulting direction the two balls head off in?  How do I apply this to each ball?     Handling the collision detection of the "walls" and the resulting vector changes were easy but I see more complications with ball-ball collisions.  With walls I simply had to take the negative of the appropriate x or y vector and off it would go in the correct direction.  With balls I don't think it is that way.  Some quick clarifications: for simplicity I'm ok with a perfectly elastic collision for now, also all my balls have the same mass right now, but I might change that in the future.    Edit: Resources I have found useful  2d Ball physics with vectors:  2d Ball collision detection example:     Success!  I have the ball collision detection and response working great!  Relevant code:  Collision Detection:  for (int i = 0; i &lt; ballCount; i++)   {       for (int j = i + 1; j &lt; ballCount; j++)       {           if (balls[i].colliding(balls[j]))           {             balls[i].resolveCollision(balls[j]);         }     } }   This will check for collisions between every ball but skip redundant checks (if you have to check if ball 1 collides with ball 2 then you don't need to check if ball 2 collides with ball 1.  Also, it skips checking for collisions with itself).  Then, in my ball class I have my colliding() and resolveCollision() methods:  public boolean colliding(Ball ball) {     float xd = position.getX() - ball.position.getX();     float yd = position.getY() - ball.position.getY();      float sumRadius = getRadius() + ball.getRadius();     float sqrRadius = sumRadius * sumRadius;      float distSqr = (xd * xd) + (yd * yd);      if (distSqr &lt;= sqrRadius)     {         return true;     }      return false; }  public void resolveCollision(Ball ball) {     // get the mtd     Vector2d delta = (position.subtract(ball.position));     float d = delta.getLength();     // minimum translation distance to push balls apart after intersecting     Vector2d mtd = delta.multiply(((getRadius() + ball.getRadius())-d)/d);        // resolve intersection --     // inverse mass quantities     float im1 = 1 / getMass();      float im2 = 1 / ball.getMass();      // push-pull them apart based off their mass     position = position.add(mtd.multiply(im1 / (im1 + im2)));     ball.position = ball.position.subtract(mtd.multiply(im2 / (im1 + im2)));      // impact speed     Vector2d v = (this.velocity.subtract(ball.velocity));     float vn = v.dot(mtd.normalize());      // sphere intersecting but moving away from each other already     if (vn &gt; 0.0f) return;      // collision impulse     float i = (-(1.0f + Constants.restitution) * vn) / (im1 + im2);     Vector2d impulse = mtd.normalize().multiply(i);      // change in momentum     this.velocity = this.velocity.add(impulse.multiply(im1));     ball.velocity = ball.velocity.subtract(impulse.multiply(im2));  }   Source Code:     If anyone has some suggestions for how to improve this basic physics simulator let me know!  One thing I have yet to add is angular momentum so the balls will roll more realistically.  Any other suggestions?  Leave a comment!