Why does continuous linear functional $f$ in topological space $V$ have to have an inverse?  Like in the proof for kernel of continuous functional being closed.       If f is continuous then $\ker f=f^{-1}(0)$ ...   But how does one know $f^{-1}$ exists?
My proof is very different from my reference, hence I am wondering is I got this right?  Apparently, $F$ is continuous, and the identity matrix is closed. Now we want to show that the preimage of continuous function on closed set is closed.  Let $D$ be a closed set, Consider a sequence $x_n \to x_0$ in which $x_n \in f^{-1}(D)$, and we will show that $x_0 \in f^{-1}(D)$.  Since $f$ is continuous, we have a convergent sequence $$\lim_{n\to \infty} f(x_n) = f(x_0) = y.$$  But we know $y$ is in the range, hence, $x_0$ is in the domain. So the preimage is also closed since it contains all the limit points.  Thank you.
Typing strange letters¿ w/o numpad? How do i type ♣ (alt+5) and ♥ (alt+259) and other characters like it on a keyboard w/o a numberpad? using the numbers 1-9-0 doesnt activate those characters. Note i want to TYPE and not use that windows program to select/copy/paste Empty drives show 2 GB used, Explorer shows nothing I have 4 drives in Windows: C:, D:, E: and F:. Except C:, all of them are empty.    Suddenly it shows up to 2 GB space used – even when there are no files.     [  When I format the drives, it's fine for a while until it shows used space again.  I use Windows 8.1 Pro 64 Bit. Showing $\frac{x}{1+x}0$ using the mean value theorem I've been living in Ireland for almost a year now and I start noticing they use the word "mens" a lot. I can see it used in:   Shops, to denote the area where you can find men's clothes In sport, when they talk about "mens team".    My guess is just that they are lazy about the use of quotes, so that mens should actually be men's. However, there may be some rule I'm not aware of. So, is "mens" only limited to Irish English? When I'm allowed to use it? Center of a free product Is it possible to have test error lower than training error?  I have a classification problem with 2000 samples, 500 of which are positives, 1500 are negatives. I split my data into 70% training data, 30% test data.  Run random forest with 200 estimators and cv=10. I did this several times and compared the recall and precision score and notice the scores for my test set is significantly better. Is this possible? How do I log all input and output in a terminal session? How do I keep logs of all the work done via terminal? For example, I want a record of all the work done on database via terminal. In Ubuntu 17.10 ubuntu upgraded Nautilus (File Browser) to version 3.26.0. New nautilus is not using file-roller anymore, it has  to an integrated compression mechanism.     This makes it complicated to create encrypted archives and/or different types compression archives that file-roller does provide.  How to make nautilus use file-roller as before?  Related question: php SimpleXML check if a child exists How do you correctly use a meat thermometer? When I add a mirror modifier it sometimes doesn't mirror correctly, giving me an overlapping mesh. How can I fix this? Solving recurrence relation. Recurrence Why do all Winforms programs require the [STAThread] attribute? Kernel did not upgrade between 14.04 LTS to 16.04 LTS Every time I think I have the curve modifier mastered, it throws me a another curve ball (no pun intended).  Why in  can I not get the lights to follow the spiral curve? When the modifier is activated, the lights are stretched and distorted along the curve.  I created the curve by converting it from a mesh (string of verts). Why does it not display like a normal Bezier? Is that related to my problem?  I am following all the "rules" that I am aware of, as detailed here: . The curve seems to run in the correct direction. The origin point is at the first point of the curve. The curve modifier on the lights is set to -Y which is the local orientation of the curve's direction.  What am I missing? In Ubuntu 18.04, when moving the windows there is a resize effect that occurs when accidentally touching one of the screen edges.  This control was available in CCSM and Ubuntu Tweak in previous versions with Unity.  I'm not finding it in GNOME Tweaks.  I also searched the repository with:  apt search ubuntu.*tweak Which serial protocol or encoding scheme does this scoreboard use? I am trying to decode a signal stream coming from a Nevco scoreboard controller:    Packets like this are sent every ~2.7 ms.  I suspect it is some sort of UART signal because it is on the RX line of a ATMEGA48, but I cannot identify the finer protocol details like parity and stop bits. All the combinations I have tried returns junk data, so it may be another encoding scheme.  What's unusual is that the same signal is also sent to the MOSI pin of the microcontroller, but there is no clock signal anywhere.  A .vcd file is .   The shortest pulses (both high and low) are always 8us in length.    This is what two packets look like. Anyone got ideas? Delete files except n files with the "bigger" name? Can I use multiple regression when I have mixed categorical and continuous predictors? It looks like you can use coding for one categorical variable, but I have two categorical and one continuous predictor variable. Can i use multiple regression for this in SPSS and if so how? thanks! pgfplots axis scaling Is it possible to define in pgfplots that y should correspond to Y cm in my plot?   I know how to do this in tkz-fct, so let me add an tkz-fct example to make more clear what I want:  \documentclass{article}  \usepackage{pgfplots} \usepackage{tkz-fct} \usetkzobj{all}  \begin{document}  \begin{tikzpicture}[scale=.5] \tkzInit[xmax=10,ystep=3,ymax=10] \tkzAxeXY \tkzGrid \tkzFct[color=red, thick,domain=0:10]{\x} \tkzFct[domain=0:10]{\x**2} \end{tikzpicture} \begin{tikzpicture}[scale=.5] \tkzInit[xmax=10,ystep=1.5,ymax=10] \tkzAxeXY \tkzGrid \tkzFct[color=red, thick,domain=0:10]{\x} \tkzFct[domain=0:10]{\x**2} \end{tikzpicture}   \end{document}     In one case 3 corresponds to 0.5 cm on the y-axis in the picture, in the other case 1.5 corresponds to 0.5 cm on the y axis. This is achieved by using the scale=0.5 option together with the ystep option.  Not that it is important for me to have grids as in the tkz-fct example.