Show that $(e_1,e_2,...)$ is not a Schauder basis of $\ell^\infty$ where $e_i$ is the vector in $\mathbb R^\infty$ with 1 in the ith coordinate and 0 elsewhere and $\ell^\infty=\{(x_1,x_2,...)|x_i\in \mathbb R ~~and ~~ \|x\|_\infty&lt;\infty\} ,\|x\|_\infty=\sup\{|x_1|,|x_2|,...\}$  I am wondering how to prove it and why this statement is true. Actually I think it is a basis since for any given vector $(x_1,x_2,...)=x_1e_1+x_2e_2+...$.  I already proved that $(e_1,e_2,...)$ is a Schauder basis of $\ell^p$ where p>1. I just can't see what changes when it comes to the case of $\ell^\infty$
Let $(e_n)$ (where $ e_n $ has a 1 in the $n$-th place and zeros otherwise) be unit  standard vectors of $\ell_\infty$.   Why is $(e_n)$ not a basis for $\ell_\infty$?  Thanks.
So, I mount a 200mm lens on my Canon 450D. It effectively becomes a 320mm lens. Is this the equivalent of 320mm on a full frame camera? That is, from what I've figured out I get an equivalent field of view but nothing I've read is indicating that I get the magnification to go along with it.  So as my question says in the title, does my crop sensor camera really turn my lens into a longer one (in terms of magnification), or does it just look like it based on the reduced field of view I get? Are the known exoplanets representative of the population of exoplanets? An  states that most of the 715 newly announced exoplanets orbit very close to their host stars.   Do the theories predict that this is a representative sample and that most planets are very close to their host stars? Or is it just that our current detection techniques are predisposed to finding only those planets close to their stars (because the sensors look for dimming of the host star for example)? All matrices which commute with all $2\times 2$ matrices I would like to find all matrices which commute with all $2\times2$ matrices.  I started solving problem in this way:    1) I have this matrix $A$ with real numbers: $$A=\left[\begin{array}{cc}a &amp;b\\c &amp;d\end{array}\right]$$ 2) Matrix which commute with matrix $A$ is matrix $B$: $$B = \left[\begin{array}{cc}e&amp; f\\ g &amp;h\end{array}\right]$$ 3) When i solve the equation $AB=BA$, I get this: $$\left[\begin{array}{cc}ae+bg&amp;af+bh\\ce+dg&amp;cf+dh\end{array}\right]=\left[\begin{array}{cc}ea+cf&amp;eb+df\\ga+ch&amp;gb+dh\end{array}\right]$$  How I can get the general look of wanted matrix? link new object to scene with python in 2.8 In Python 2.8, I would like to create copies of the object 'CubeTemplate' but am having no luck.  I have reviewed every blender.stackexchange post on this subject (, ,  and ) but cannot get them to work - either they don't apply or the sample code throws errors (perhaps because it was pre 2.8).  The code below throws this error:       AttributeError: 'bpy_prop_collection' object has no attribute 'link'   for x in range(1, 3):     bpy.data.objects['CubeTemplate'].select_set(True)     bpy.context.view_layer.objects.active = bpy.data.objects['CubeTemplate']     me = bpy.context.object.data # use current object's data     me_copy = me.copy()     ob = bpy.data.objects.new('CubeCopy'+str(x), me_copy)     ob.location = (0, 0, x)     scene = bpy.context.scene     scene.objects.link(ob)     scene.update() Why is the picture not inserted? When I pdflatex the text below it doesn't insert the picture, but writes the name of the file instead (pic).  And the geometry line gives the error  ! LaTeX Error: Missing \begin{document}.   The picture I am using is at  and the floatrow package .  Question  Can anyone see what is wrong?  Text  \documentclass[a4paper]{article} \pdfpagewidth=\paperwidth \pdfpageheight=\paperheight \usepackage{aeguill} \pdfcompresslevel=9 \usepackage[pdftex,colorlinks]{hyperref}  \usepackage{a4} \usepackage{fixltx2e} \usepackage[danish]{babel} \usepackage[latin1]{inputenc} \usepackage[T1]{fontenc}  \usepackage[draft]{graphicx} \usepackage{floatrow}  \usepackage{geometry}  \geometry{margin=10pt,paperwidth=\textwidth+20pt,paperheight=\textheight+20pt}  \begin{document}  The first paragraph.  \begin{figure}   \includegraphics{pic} \end{figure}  And the second.  \end{document} Cardinality of set of real continuous functions I found several plugins (like  and ) to get the ability to attach a picture to a WP user. But these plugins are not maintained any more.  Since there's apparently no plugin that does it today, is there an easy way to do it manually? Every time I go to install a new app, I get the message "Insufficient storage", and an unable to install anything!  I am using Samsung GT19060 my internal memory is 8GB. I have not downloaded anything but anytime I try to download something it will tell me low memory space. As a developer of free software, I want to get my application into Ubuntu.  Should I provide a source tarball and ask for packaging? Or should I create a package myself? Should I target Ubuntu directly, or should I try to get into Debian and then into Ubuntu automatically? Is there a word for four times as much, analogous to once, twice, and thrice? I am using \tableofcontents to create TOC in my document. however it is appearing as shown in the image. As the numbers increase the gap between the number and text is reducing.   Thanks for your help Aku Meta Stack Exchange should not penalize downvotes Due to my ignorance, I find that most of the references for mathematical analysis (real analysis or advanced calculus) I have read do not talk much about the "multivariate calculus". After dealing with the single variable calculus theoretically, it usually directly goes to the topic of measure theory.    After reading the wiki article "", I'd like to find the rigorous proof for this test. The first book comes to my mind is Courant's Introduction to Calculus and Analysis which includes the multivariate case in the second volume.   Motivated by this, I'd like to put the question here:     What are the usual references for the   theoretical treatment for   ? Show us who submitted offensive and comment flags Find Neural Network Inputs Given Outputs I've trained a neural network with two inputs, a single hidden layer with two neurons, and one output using a bipolar sigmoid activation function. If a single input is known, how would I determine the second input to create a desired output?  For example, let's say the neural network is trained to add two inputs to produce an output. So if input_1 = 3 and input_2 = 4, the output will be 7, (3 + 4 = 7). Given input_1 = 3 and the desired output is 7, I want to calculate the second input required to produce the desired output (the answer should be 4).  How would I do this for a network that is more complicated than basic addition and has multiple inputs/outputs? For example, for a network with four inputs and two outputs, how would I calculate input_3 and input_4 given input_1, input_2, output_1 and output_2? The question is as below:      Show that each vertex $v$ of a tree $T$ appears $deg(v)-1$ times in the Prüfer encoding of $T$, where $deg(v)$ is the degree of $v$ in $T$.   For this, I think there are 3 scenarios a leaf should be considered for: (1) when it is a leaf as part of a tree; (2) a leaf that has been discarded; (3) the remaining edge. I feel like this will result in a short proof which is only two (or so) statements long, but at the same time, feel it is too simple.   EDIT: I've had a look at this again and narrowed it down to 2 cases:   Case 1: The vertex $v$ in question has $deg(v)\ge2$. $v$ then loses its neighbours as the algorithm is applied repeatedly until $v$ becomes a leaf itseelf. At this point, $v$ is attached to another neighbour, leaving an edge. This means the algorithm can no longer be applied and we have obtained the final Prüfer Encoding for the tree $T$.  Case 2: The vertex $v$ has $deg(v)\ge 2$. It loses its neighbours as the algorithm is applied repeatedly but is deleted itself at some point in the process.   Have I gone about this wrong?  Thank you. HSA - Continue to Invest? I remember reading a sci-fi story where astronauts visit a planet, which is covered in stationary aliens with mile long necks that filter feed from the atmosphere.  It was published between 1960 and 2000.    It was in English and involved a pair of astronauts that went from planet to planet on a tour or something, though I don't know details of any other planets they visited. 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. Some common problems with displaying routes created from street data, such as bus routes, include:   Routes may overlap themselves, such as when a route takes the same road in opposite directions Routes may overlap other routes that take the same street. Routes are often overly detailed, or "noisy", owing to their derivation from street data. Indicating a route's direction unambiguously without overloading it with arrows can be a challenge.   What are some techniques for dealing with these issues in order to create a practical and usable route map, such as this ? Do you know of any such public transport maps that were produced by automatic techniques rather than manual illustration/digitization?  Specifically I am using ArcGIS 10, but solutions involving other software are welcome as well.  One technique I have had some success with for dealing with 1) is using ArcMap's  with a negative offset -- assuming the lines are oriented in the direction of travel, this will offset the lines such that the right and left-hand sides are distinct and oriented as you would expect (at least for  countries, for left-hand traffic countries one would use a positive offset). However the result tends to be a bit wonky around turns, interchanges and and U-turns, and is inflexible when dealing with multiple routes.  Another possibility that I am not very familiar with but which seems promising is  -- however, I have not had much luck in finding examples of using representations for transportation maps and whether it might fit the bill.