I'm reading Axler "Linear agebra done right" and in Chapter 1 he discusses subspaces and direct sum. My question is, are there subspaces of the infinite-dimensional vector spaces, e.g. a functional Banach space with sup norm $V$ that directly sum to the entire space $$V = U_1\oplus U_2$$ (except for trivial $U_1=\{0\}$, $U_2 = V$).
I am trying to prove that all vector spaces can be written as the direct sum of two subspaces.  My attempt: If each vector in vector space V can be uniquely written as the sum of two other vectors in V, the two vectors can be categorized into subspaces and thus V can be written as the direct sum of these subspaces. I am not sure how to show that this is true for all cases though. Edit Sorry for the confusion,to clarify I meantGiven a vector space V with a subspace W, prove that there exists a subspace W' such that the direct sum of W and W' is V.I am not sure about my proof method at all. Thank you.
Can the Next/Prev Post links be ordered by menu order or by a meta key? I have a series of posts that are ordered by a meta_key value. They could also be arranged by menu order, if necessary.  The next/prev post links (generated by next_post_link, previous_post_link, or posts_nav_link all navigate by chronology. While I understand this default behaviour, I don't understand how to change it. I found that it maps through to adjacent_post_link in link-template.php, but then it starts to seem fairly hard-coded. Is it recommended to re-write this from scratch to replace it, or is there a better solution. I pick a random subset $S$ of $\{1,\ldots,N\}$, and you have to guess what it is. After each guess $G$, I tell you the number of elements in $G \cap S$. How many guesses do you need? How can I use the MS JDBC driver with MS SQL Server 2008 Express? My configuration:     windows XP SP3 JDBC 2005 MS SQL Server 2008 Express, exposed via tcp/ip on port 1433 sqljdbc.jar in class path    I tried:  try {     Class.forName("com.microsoft.sqlserver.jdbc.SQLServerDriver").newInstance();     con = DriverManager.getConnection("jdbc:microsoft:sqlserver://localhost:1433/SQLEXPRESS2008;databaseName=Test;selectMethod=cursor", "sa", ""); } catch (Exception e) {     e.printStackTrace(); }   But it always throws an exception:  java.sql.SQLException: No suitable driver   I also tried the following urls:  localhost:1433/SQLEXPRESS2008  localhost/SQLEXPRESS2008  localhost   Same results.  Any help? Installing drivers for windows? How can I prevent data theft from a company?  Are there any solutions to this problem?  Is there any software that encrypts important data (.doc, .xls, etc.) and prevents it from copying?  I need a solution where employees can read/write documents but can not copy then to a USB drive or e-mail them. Lets say we have a dependent variable $Y$ with few categories and set of independent variables.    What are the advantages of multinomial logistic regression over set of binary logistic regressions (i.e. )? By set of binary logistic regression I mean that for each category $y_{i} \in Y$ we build separate binary logistic regression model with target=1 when $Y=y_{i}$ and 0 otherwise. Why is integration the inverse of differentiation, I mean why do I get the same function when I integrate and then differentiate the result? jQuery Datepicker with text input that doesn't allow user input How do I use the jQuery Datepicker with a textbox input:  $("#my_txtbox").datepicker({   // options });   that doesn't allow the user to input random text in the textbox. I want the Datepicker to pop up when the textbox gains focus or the user clicks on it, but I want the textbox to ignore any user input using the keyboard (copy &amp; paste, or any other). I want to fill the textbox exclusively from the Datepicker calendar.  Is this possible?  jQuery 1.2.6 Datepicker 1.5.2 Obtaining knowledge from a random forest Random forests are considered to be black boxes, but recently I was thinking what knowledge can be obtained from a random forest?  The most obvious thing is the importance of the variables, in the simplest variant it can be done just by calculating the number of occurrences of a variable. The second thing I was thinking about are interactions. I think that if the number of trees is sufficiently large then the number of occurrences of pairs of variables can be tested (something like chi square independence).  The third thing are nonlinearities of variables. My first idea was just to look at a chart of a variable  Vs score, but I'm not sure yet whether it makes any sense.    Added 23.01.2012 Motivation    I want to use this knowledge to improve a logit model. I think (or at least I hope) that it is possible to find interactions and nonlinearities that were overlooked. When is $\mathbb{F}_p[x]/(x^2-2)\simeq\mathbb{F}_p[x]/(x^2-3)$ for small primes? Why can I put an electrolytic capacitor on AC? I have seen here and there schematics with electrolytic capacitors put on AC. And this sounds weird to me.  Electrolytic capacitors have a polarity, right? If we invert the polarities on DC, bad things happen. As far as I understand, AC inverts polarity every now and then (commonly 50Hz). Why can we put such capacitors on AC without damaging them?  Example:    from demonstration here: What books, journals, and electronic resources are most valuable for expanding knowledge of GIS? What books, journals, and/or electronic resources have you found most valuable for expanding your knowledge in the GIS field, and why? On my homepage, I'm using this method to hide my email from spam bots:  &lt;a href="admin [at] example.com"    rel="nofollow"    onclick="this.href='mailto:' + 'admin' + '@' + 'example.com'"&gt;Contact me&lt;/a&gt;   What do you think about it? Is it effective? What other methods do you know or use? How to use Terminal to delete all .svn folders recursively? What command can I type into the Terminal so that I can delete all .svn folders within a folder (and from all subdirectories) but not delete anything else? Is the Doctor Who episode 'Blink' the first use of the phrase "timey-wimey"? What are the biggest changes between UA Ranger and PHB Ranger? Derive a formula to find the number of trailing zeroes in $n!$ Why does sstatic.net serve cookies? According to :  This is a fast, cookieless domain intended for static content delivered to the   However, all responses from the server set a cookie anyway: $ http https://sstatic.net/ HTTP/1.1 200 OK CF-RAY: 162b88245bc406be-LHR Cache-Control: max-age=604800 Connection: keep-alive Content-Encoding: gzip Content-Type: text/html Date: Sun, 31 Aug 2014 19:11:04 GMT ETag: W/&quot;d5474fdfb5c3cf1:0&quot; Last-Modified: Fri, 29 Aug 2014 18:20:12 GMT Server: cloudflare-nginx Set-Cookie: __cfduid=db60cc6f3e7f6482e37f1f2b85c9dc5881409512264375; expires=Mon, 23-Dec-2019 23:50:00 GMT; path=/; domain=.sstatic.net; HttpOnly Transfer-Encoding: chunked Vary: Accept-Encoding [...]  Is this intentional, or just funny? Edited to add: I understand that the cookie is used to track visiting browsers so that cloudflare can distinguish them from misbehaving hosts using the same or a nearby IP address. However, if the policy is intentionally to serve cookies, then the self-documentation of sstatic.net as a &quot;cookieless domain&quot; is incorrect. multiline command chunks in bash history into multiple lines In Bash (as well as Bash in WSL), multiline written commands can call-back with up/down arrows correctly also built-in history command report them correctly (until I'm still logged-in in bash). But after exiting from bash and press up/down arrows, multiline commands appears each as separate entries and that's because cat ~/.bash_history has chucked the command on multiple separate entries and writing problem here; the history command result messed too. I have set these options in ~/.bashrc shopt -s cmdhist lithist HISTTIMEFORMAT='%F %T '  and now writing problem fixed but history command still report in multiple line and separate entries (reading problem now). Bash version is 4.3, same issue on v4.4 there is  since a long years but no solution yet. Update: it's also to mention that below multiline command as an example is correctly displaying even after exit Bash: for i in {1..10} do echo $i done  but this is failing: awk ' { print $1}' &lt;&lt;&lt;'first last'  Update 2: the issue mentioned above is fixed in Bash v5.0.0+ that changed to recognize entries between &quot;timestamp as delimiter&quot; as single entry of command (HISTTIMEFORMAT, cmdhist and lithist must be set). I remember that the main character is trying to find the group that destroyed his home planet while he was on a mission. His planet was very difficult to live on and the people became expert in martial arts and were the best soldiers in the galaxy. They were mercenaries and often helped against tyrants. There was a very memorable section in at least one book about an initiation task that the main character had to do in that he had to survive in their wilderness for days. He was at some point surrounded in a old building by thousands of poisonous worm/snake creatures that he managed to survive until rescued by the leaders.