I would like to show that a bounded linear operator $T: \ell^2(\mathbb{N}) \rightarrow \ell^1(\mathbb{N})$ cannot be surjective.  If we assume surjectivity, then by the Open Mapping Theorem, $T$ is an open map, i.e. for every open $O \subset \ell^2(\mathbb{N})$, its image $T(O)$ is open in $\ell^1(\mathbb{N})$.   Is it then possible to contradict Baire's Theorem by constructing a sequence of open dense subsets of $\ell^1(\mathbb{N})$, whose intersection is not dense?  Any help pointing me into the right direction would be appreciated.
The second part of an exercise asks that I show:     There does not exist a surjective bounded linear map from $\ell^2(\mathbf{N})$ to $\ell^1(\mathbf{N})$.     The first part was to show that these two spaces are separable in their respective $L^p$ norms.    I've really hit a wall on this one.  I know, in fact, that there does exist a surjective bounded linear map from $\ell^1(\mathbf{N})$ to $\ell^2(\mathbf{N})$.  So my intuition says that if there were also one going the other way, then the two spaces would have to be isomorphic, which is false .  But I don't know of any such theorem that I could apply to "the category of Banach spaces".    My only other idea is to argue by contradiction using the open mapping theorem, but nothing is coming to me...  My guess is that separability is also going to be important, given the first part of the exercise.  -Thanks.
How to Backup and Restore Android OS without Rooting I want to find a way to restore my phone in case the process of rooting my phone or dual booting a custom rom breaks it. However, most of the ones I've looked at require the phone to be rooted before installing. I know I can create a nandroid backup through system settings, but I don't know how to recover that without rooting my device, or if there are better options. I'm not terribly concerned for any personal data on my phone (I can backup my pictures pretty easily), but I mostly just want to be able reinstall Android OS or reset to factory default in case my device breaks. Thanks! Edit: I am going to ask a new question (one different enough to justify not simply modifying this one) with the new info I learned from the possible duplicate. I registered my Stack Overflow account using the Stack Exchange OpenID provider. I later changed my e-mail, but I cannot change the e-mail address associated with my OpenID for some reason. Why is that, and how do I change it? I logged in to Stack Exchange and it lets me change my password, but never my e-mail. How to make the space under footnotes disappear? How do I make the space under footnotes disappear? And to space over the footnote's bar?    Code:  \documentclass[12pt,french]{report} \usepackage{mathptmx} \usepackage{helvet} \usepackage{courier} \usepackage[T1]{fontenc} \usepackage[cp1250]{inputenc} \usepackage[a4paper]{geometry} \geometry{verbose,margin=1in,headheight=12pt,headsep=25pt} \usepackage{fancyhdr} \pagestyle{fancy} \setlength{\parskip}{\medskipamount} \setlength{\parindent}{0pt} \usepackage{babel} \makeatletter \addto\extrasfrench{%    \providecommand{\og}{\leavevmode\flqq~}%    \providecommand{\fg}{\ifdim\lastskip&gt;\z@\unskip\fi~\frqq}% } \makeatother \usepackage{latexsym} \usepackage{pifont} \usepackage{color} \usepackage{float} \usepackage{textcomp} \usepackage{graphicx} \usepackage{setspace} \usepackage[unicode=true,  bookmarks=true,bookmarksnumbered=true,bookmarksopen=false,  breaklinks=true,pdfborder={0 0 0},backref=false,colorlinks=false]  {hyperref} \hypersetup{pdftitle={tito},  pdfauthor={abc},  pdfkeywords={kiko}} \usepackage{lipsum}  \makeatletter \usepackage{multicol} \usepackage{fancyhdr} \usepackage{multirow} \usepackage{longtable} \setcounter{secnumdepth}{4} \setcounter{tocdepth}{4} \usepackage{array} \newcommand{\ts}{\textsuperscript} \renewcommand\theparagraph{\alph{paragraph}.} \renewcommand\paragraph{\@startsection{paragraph}{4}{\z@}%                                     {3.25ex \@plus1ex \@minus.2ex}%                                     {0.5ex \@plus .2ex}%                                     {\normalfont\normalsize\bfseries}} \renewcommand*\l@paragraph{\@dottedtocline{4}{10em}{2em}}  \def\thickhrulefill{\leavevmode \leaders \hrule height 1ex \hfill \kern \z@}  \def\@makechapterhead#1{% \vspace*{10\p@}% {\parindent \z@ {\reset@font \usefont{OT1}{phv}{m}{n} \LARGE Chapitre \thechapter\par\nobreak}% \par\nobreak \vspace*{30\p@} \interlinepenalty\@M \usefont{OT1}{ptm}{b}{n} {\raggedright \Huge #1}% \par\nobreak \vskip 20\p@ \hrule height 1pt \par\nobreak \vskip 45\p@ }}  \def\@makeschapterhead#1{% \vspace*{10\p@}% {\parindent \z@ {\raggedleft \reset@font \scshape \vphantom{\@chapapp{} \thechapter} \par\nobreak}% \par\nobreak \vspace*{30\p@} \interlinepenalty\@M \usefont{OT1}{ptm}{b}{n} {\raggedright \Huge #1}% \par\nobreak \par\nobreak \vskip 45\p@ }}  \lhead{\nouppercase\leftmark} \chead{} \rhead{} \lfoot{} \cfoot{\thepage} \rfoot{} \renewcommand{\footrulewidth}{0.4pt}  \pretolerance=10000 \hyphenpenalty=10000 \exhyphenpenalty=10000  \AtBeginDocument{   \def\labelitemi{\Pisymbol{psy}{183}}   \def\labelitemii{\(\rhd\)}   \def\labelitemiii{\(\rhd\)}   \def\labelitemiv{\(\ast\)} } \makeatother  \begin{document} \chapter{Chap} \lipsum[1]\footnote{\lipsum[1]} \lipsum[1-8] \end{document} Add folder name to beginning of filename How to handle users who don't select answers for their questions? Evaluating this integral : $ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $ The question :  $$ \int \frac {1-7\cos^2x} {\sin^7x \cos^2x} dx $$  I tried dividing by $\cos^2 x$ and splitting the fraction.  That turned out to be complicated(Atleast for me!)  How do I proceed now? I'm using the Apache Commons EqualsBuilder to build the equals method for a non-static Java inner class. For example:  import org.apache.commons.lang.builder.EqualsBuilder;  public class Foo {     public class Bar {         private Bar() {}          public Foo getMyFoo() {             return Foo.this         }          private int myInt = 0;          public boolean equals(Object o) {             if (o == null || o.getClass() != getClass) return false;              Bar other = (Bar) o;             return new EqualsBuilder()                 .append(getMyFoo(), other.getMyFoo())                 .append(myInt, other.myInt)                 .isEquals();         }     }      public Bar createBar(...) {         //sensible implementation     }      public Bar createOtherBar(...) {         //another implementation     }      public boolean equals(Object o) {         //sensible equals implementation     } }   Is there syntax by which I can refer to other's Foo reference apart from declaring the getMyFoo() method? Something like other.Foo.this (which doesn't work)? Dictionary application needs to download dictionary "Apple Dictionary" before use When trying to restore a backup to a SQL Server Express 2008 database I got the following error:  Restore failed for Server '...\SQLEXPRESS'.  (Microsoft.SqlServer.SmoExtended) System.Data.SqlClient.SqlError: The database was backed up on a server running version 10.50.1600.  That version is incompatible with this server, which is running version 10.00.2531.  Either restore the database on a server that supports the backup, or use a backup  that is compatible with this server. (Microsoft.SqlServer.Smo)   Is there a way to get a backup which is compatible with the older (in my case 10.00.2531) version from the newer (in my case 10.50.1600) version of SQL Server Express? I did a fresh install of Ubuntu 16.04. /dev/md1 contains an ext4 filesystem that is mounted at /:  /dev/md1 on / type ext4 (rw,relatime,data=ordered)   Now I want to set the automatic filesystem check interval to one month, but I'm getting a strange error that I find no useful information about:  &gt; tune2fs -i 1m /dev/md1 tune2fs 1.42.13 (17-May-2015) tune2fs: Filesystem has unsupported read-only feature(s) while trying to open /dev/md1 Couldn't find valid filesystem superblock.   Rebooting didn't help. On another machine running Ubuntu 12.04, the same command works like a charm.  Does anyone have an idea what's wrong?  The command tune2fs -l /dev/md1 works fine, by the way. Here is its output:  tune2fs 1.42.13 (17-May-2015) Filesystem volume name:   &lt;none&gt; Last mounted on:          / Filesystem UUID:          38366a40-bbe1-476b-acfe-d271be6335eb Filesystem magic number:  0xEF53 Filesystem revision #:    1 (dynamic) Filesystem features:      has_journal ext_attr resize_inode dir_index filetype needs_recovery extent 64bit flex_bg sparse_super large_file huge_file dir_nlink extra_isize metadata_csum Filesystem flags:         signed_directory_hash Default mount options:    user_xattr acl Filesystem state:         clean Errors behavior:          Continue Filesystem OS type:       Linux Inode count:              8380416 Block count:              33521664 Reserved block count:     1676083 Free blocks:              32419430 Free inodes:              8312762 First block:              0 Block size:               4096 Fragment size:            4096 Group descriptor size:    64 Reserved GDT blocks:      1024 Blocks per group:         32768 Fragments per group:      32768 Inodes per group:         8192 Inode blocks per group:   512 Flex block group size:    16 Filesystem created:       Thu May 31 10:01:27 2018 Last mount time:          Thu May 31 10:49:41 2018 Last write time:          Thu May 31 10:15:14 2018 Mount count:              7 Maximum mount count:      -1 Last checked:             Thu May 31 10:01:27 2018 Check interval:           0 (&lt;none&gt;) Lifetime writes:          5782 MB Reserved blocks uid:      0 (user root) Reserved blocks gid:      0 (group root) First inode:              11 Inode size:               256 Required extra isize:     32 Desired extra isize:      32 Journal inode:            8 First orphan inode:       4325378 Default directory hash:   half_md4 Directory Hash Seed:      9d55d989-2c2b-4ad8-a29e-53c8798ec68e Journal backup:           inode blocks Checksum:                 0x3d9f8d23 What steps can I take to try to recover lost or inaccessible data from any storage device?  Answers:   This applies to any computer storage device, e.g. internal/external hard drives, USB sticks, flash memory. The most important thing is to STOP using it, any type of I/O can ruin your chances of a recovery.     We have separate questions covering common problems with USB flash drives in greater detail: Disk full, du tells different. How to further investigate? I have a SCSI disk in a server (hardware Raid 1), 32G, ext3 filesytem. df tells me that the disk is 100% full. If I delete 1G this is correctly shown.  However, if I run a du -h -x / then du tells me that only 12G are used (I use -x because of some Samba mounts).  So my question is not about subtle differences between the du and df commands but about how I can find out what causes this huge difference?  I rebooted the machine for a fsck that went w/out errors. Should I run badblocks? lsof shows me no open deleted files, lost+found is empty and there is no obvious warn/err/fail statement in the messages file.  Feel free to ask for further details of the setup. If friction acts on car all time, why is it not accelerating forever? In the first gear, slowly releasing my clutch i push the gas pedal to the maximum torque level, say at 2000 rpm. Now as the engine torque is acting on the wheel, friction helps the car to accelerate.   If i hold the gas pedal at that position, the torque should continuously act on the wheel and the car should increase speed forever. But in reality even if we put the gas pedal at the max.torque rpm range in first gear it speeds up to certain wheel rpm (say 500rpm) and it stays there only. Why? I need help with this exercise.  I need to prove $$\int_{0}^{1}x^{-x}\ dx=\sum_{n=1}^{\infty}n^{-n}$$  I think I should use some convergence theorem, but I'm stuck.  Thanks a lot! How to delete duplicate LiDAR points? How can I delete duplicate LiDAR points? I incorrectly read the same las twice and saved as a one file resulting in duplicate XY points.  How should I delete these duplicate points? Got French Schengen visa but no "VIS" after giving fingerprints I had my French Schengen visa interview 3 days ago (got the stamped visa and passport within two days, super efficient!). However there's no "VIS" written on the visa, even though I gave my fingerprints during the interview in US. I had presumed VIS would be written on the visa since I gave my fingerprints, or is it for something else? I do remember it took a while to take the fingerprints because the machine looked old and was taking a long time to read my fingerprints.  The next time I want to apply for a visa, do I need to go for an interview again just because the visa has no VIS on it? Why does change in speed of a wave make it refract? When a light wave enters a medium with a higher refractive index (e.g. from air to standard glass) and its speed decreases, why does that make it refract/bend?  I understand that wavelength decreases and frequency stays the same and therefore its speed decreases, but I can't find anywhere whatsoever why the speed decrease cause the wave to refract. So could someone please explain this? How can I get bash to perform tab-completion for my aliases? I have a bunch of bash completion scripts set up (mostly using bash-it and some manually setup).    I also have a bunch of aliases setup for common tasks like gco for git checkout. Right now I can type git checkout dTab and develop is completed for me but when I type gco dTab it does not complete.    I'm assuming this is because the completion script is completing on git and it fails to see gco.    Is there a way to generically/programmatically get all of my completion scripts to work with my aliases?  Not being able to complete when using the alias kind of defeats the purpose of the alias. When a lattice is a lattice of open sets of some topological space? When a lattice $(L,\leqslant)$ is a lattice of open (or closed) sets of some topological space $(X,\tau)$? Which conditions have to be satisfied? We may assume that $X$ is $T_1$. Pretty much the problem asks if a function is continuous at the point $c$ and $f(c) &gt; 0$ then there exists a $d &gt; 0$ such that $\forall x$, $f(x) &gt; 0$ with $|x-c| &lt; d$.   I can understand what the problem means. That if a function is positive at a point then there is another point that's really close that'll also be positive. I can not prove this in a formal way though. I've tried using the intermediate value theorem and I do not know hot to implement it. Any ideas?