Let $A_{(n)}:\mathcal B\rightarrow \mathcal B$ be a linear bounded operators such that for any $f\in \mathcal B$, $A_nf\rightarrow Af$. Show that then for any compact operator $B$  $$\|A_nB-AB\|\rightarrow0$$  Any tips how to show this?
$A_n:Y\rightarrow Z$ are operators that strongly converge to $A$. Also, $\|A_n\|_\text {op}\le c$ for $c&gt;0$. Given a compact operator $T:X\rightarrow Y$, I need to show that $A_nT$ converges to $AT$ in operator norm (all spaces in this question are Banach spaces).     I was unable to prove this and I also do not understand why we need to assume that $T$ is compact. Any ideas?
What is the proper way not to scare civilians? Writing a git post-receive hook to deal with a specific branch Difference between logit and probit models Notification that ‘someone might be watching what you browse’ I got a notification on my iPhone. The notification reads     Someone might be watching what you browse. Follow 3 steps to encrypt your web traffic and change your IP address.   What does this mean? Create element in pdf that doesn't print to paper In an older question from June 2011, , apparently revived by the system, a user is asking about the origin of the rotation of celestial bodies.  The  (with 17 votes) states the following:     When things collapse under their own gravity in space (i.e. clouds of gas and dust), any small amount of asymmetry in the collapse will be enough start it spinning.   And then it goes on saying:     Even if it spins by a tiny amount, as it collapses, angular momentum     conservation will mean it spins more and more quickly.   Is all this really correct?  I could try to ask in a comment to the answer. I actually did. But the author is no longer registered.  Hence I am asking it as a question.  But that is not the end of the story. As I was about to ask my question, I looked for other relevant questions, and there are some. I did not make an exaustive survey, but I did not find one that had satisfactory answers:    (december 2011) has just one accepted answer (0 votes) stating essentially that the solar system has angular momentum because it started from a cloud that had angular momentum. No comment.  (july 2011) is questionning about everything from galaxies to electrons. The answers are of a much higher quality, but while they seem to justify the possiblility of spin and angular momentum (and its conservation), there are not clear on how it appears, at least for celestial bodies.  (june 2013): That was my own very naive first question on this site. It is related but does not directly address spin.   One answer I found is that momentum appear from bodies in relative motion past each other. Nice, but not really an answer: where did they get their relative speed to begin with. Unfortunately the answer is: from the angular momentum of larger structures they belong to (well, it can be more comples). So this is just chicken and egg explanation. But how did it get started?  Note that my question is not a duplicate, since it is primarily whether the above statements are correct, which is not considered in these other questions.  Furthermore, these questions are fairly old, and given that none had received answers that seem to answer the question of the origine of existence of celestial angular momentum (not in a way I can understand anyway), it may be appropriate to attempt restarting the process, with the above discussion to motivate that an answer should explain the physical mechanism that started the existence of spin of celestial bodies. How can I highlight text before a character with listing? All is in the title. I would like to add a color to the text before a character with listing package.  For example, I would like to add a color to text between the beginning of a line and : or =, in a makefile.  I tried to use moredelimiters but I did not succeed.    \documentclass{article}  \usepackage{listings} \usepackage{xcolor} \definecolor{keycolor}{RGB}{172, 42, 42} \definecolor{mbleu}{RGB}{64,96,127} \definecolor{vimvert}{RGB}{46, 139, 87}  \lstdefinestyle{global}{   basicstyle=\ttfamily\scriptsize\color{black!90},%   stringstyle=\itshape\color{magenta},%   showstringspaces=false,%   keywordstyle={\bfseries\color{keycolor}},%   commentstyle=\color{blue}\slshape,%   framexleftmargin=1mm,%   backgroundcolor=\color{black!2},% }  \lstdefinestyle{makefile}{   style=global,%   morecomment=[l][commentstyle]{\#},%   emphstyle={\color{vimvert}},%   moredelim=[s][emphstyle]{\$(}{)}% }  \begin{document}  \begin{lstlisting}[style=makefile] .SUFFIXES: .inc .f .f90 .F  # all CPP processed fortran files have the extension .f90 SUFFIX=.f90  #----------------------------------------------------------------------- # this release should be fpp clean # we now recommend fpp as preprocessor # if this fails go back to cpp CPP_=fpp -f_com=no -free -w0 $*.F $*$(SUFFIX)  vasp: $(SOURCE) $(FFT3D) $(INC) main.o      rm -f vasp     $(FCL) -o vasp main.o  $(SOURCE)   $(FFT3D) $(LIB) $(LINK) makeparam: $(SOURCE) $(FFT3D) makeparam.o main.F $(INC)     $(FCL) -o makeparam  $(LINK) makeparam.o $(SOURCE) $(FFT3D) $(LIB) zgemmtest: zgemmtest.o base.o random.o $(INC)     $(FCL) -o zgemmtest $(LINK) zgemmtest.o random.o base.o $(LIB) dgemmtest: dgemmtest.o base.o random.o $(INC)     $(FCL) -o dgemmtest $(LINK) dgemmtest.o random.o base.o $(LIB)  ffttest: base.o smart_allocate.o mpi.o mgrid.o random.o ffttest.o $(FFT3D) $(INC)     $(FCL) -o ffttest $(LINK) ffttest.o mpi.o mgrid.o random.o smart_allocate.o base.o $(FFT3D) $(LIB) kpoints: $(SOURCE) $(FFT3D) makekpoints.o main.F $(INC)     $(FCL) -o kpoints $(LINK) makekpoints.o $(SOURCE) $(FFT3D) $(LIB)  clean:       -rm -f *.g *.f *.o *.L *.mod ; touch *.F \end{lstlisting}  \end{document} How can I download an entire website? Can not update Bash on Debian 6.0 (Squeeze) I can not update  on a Debian 6.0 (Squeeze) server to get rid of the discovered vulnerability:  bash --version GNU bash, version 4.1.5(1)-release (x86_64-pc-linux-gnu)  apt-get update apt-get install bash Reading package lists... Done Building dependency tree Reading state information... Done bash is already the newest version. 0 upgraded, 0 newly installed, 0 to remove and 7 not upgraded.   Can I use Squeeze-LTS for this server just to update Bash? After one week I will be on another server, so I will not make any other updates.  uname -m x86_64  lsb_release -a No LSB modules are available. Distributor ID: Debian Description:    Debian GNU/Linux 6.0.5 (squeeze) Release:        6.0.5 Codename:       squeeze given that $f$ is an analytic function with real part $u$ is a polynomial in the variable of $x,y$, $z=x+iy$, we need to show $f$ is a polynomial in $z$, I am kind of puzzled to see the problem, first I would like to say:any hint please The extension of smooth function Asymptotic behavior of a trigonometric series Consider the following series $$S_N = \sum_{k=1}^N \frac{\sin[k\pi/(N+1)]}{k}.$$ What is the asymptotic behavior of this series for $N \to \infty$?  If we replace the sine function with the factor $1$, we get the standard harmonic series which diverges $\sim \ln N$. However, if we replace the sine function with $(-1)^k$, we get a convergent series with the result $\ln 2$. Therefore, I think that $S_N$, defined above, should diverge, but more slowly than the logarithm. Am I correct? Remove border from IFrame How would I remove the border from an iframe embedded in my web app? An example of the iframe is:  &lt;iframe src="myURL" width="300" height="300"&gt;Browser not compatible.&lt;/iframe&gt;   I would like the transition from the content on my page to the contents of the iframe to be seamless, assuming the background colors are consistent. The target browser is IE6 only and unfortunately solutions for others will not help. Could normal users have an easier way to flag for migration? Shapefile layer no longer editable (toggle edit option greyed out) in QGIS? How can I serialize an object to JSON without properties that are null? To communicate with a REST interface on a private system, I need to send a JSON message from a trigger. I have a message object that has some fields, lists and a child object:  public class MerchantConfig {  public class MIDConfig {     public String mid;     public String currencyCode;     public String cardTemplateName;     public String paymentPanelRestriction;     public List&lt;String&gt; restrictedCountries;      public String paymentPanelRestrictionForCountries; }  public String merchantName; public String sopgTemplateName; public Integer dispositionTimeWindowInMinutes; public List&lt;String&gt; features; public Boolean activateImmediately; public List&lt;MIDConfig&gt; midConfiguration; }   I can serialize it to JSON using the built-in serializer:  String body = JSON.serializePretty([MerchantConfig instance]);   The result:  {   "sopgTemplateName" : null,   "midConfiguration" : [ {     "restrictionCountries" : [ "DE", "AT" ],     "paymentPanelRestrictionForCountries" : "MANDATORY",     "paymentPanelRestriction" : "OPTIONAL",     "mid" : "1000000001",     "currencyCode" : "EUR",     "cardTemplateName" : null   }, {     "restrictionCountries" : [ ],     "paymentPanelRestrictionForCountries" : "",     "paymentPanelRestriction" : "OPTIONAL",     "mid" : "1000000002",     "currencyCode" : "USD",     "cardTemplateName" : null   }, {     "restrictionCountries" : [ ],     "paymentPanelRestrictionForCountries" : "",     "paymentPanelRestriction" : "PROHIBITED",     "mid" : "1000000003",     "currencyCode" : "AUD",     "cardTemplateName" : null   } ],   "merchantName" : "Test Merchant",   "features" : [ XYZ ],   "dispositionTimeWindowInMinutes" : 10,   "activateImmediately" : true }   How can I serialize this object so that all properties that are null don't show up in the JSON string?   In this example, "sopgTemplateName" and all "cardTemplateName" properties are null and should not appear in the JSON string. The empty list "restrictionCountries" should still appear as an empty list, though. Fitting an infinite collection of balls in an infinite dimensional unit ball Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius $\frac{1}{4}$ in the unit ball?  I guess one can immediately reduce the problem to a normed linear space of countably infinite dimensions. The solution seems clear if the concept of orthogonality exists, but not every normed linear space has an inner product so it's not possible to apply something like Gram-Schmidt to produce an orthogonal basis. Is there any way around this, or there another approach that can be used? Labeling Multiple Equations Per Row of a Table Compile nested Sums word choice - realize notice find out figure out