Let X be an infinite dimensional  normed $\mathbb K$-linear space .Prove that $\partial B(0,1)=\{x\in X: ||x||= 1\}$ is not weakly closed in $X$.   We know that a convex subset of $X$ is weakly closed iff it is closed w.r.t. the normed topology. But here $\partial B(0,1)$ is not convex in X.   (definition of weak topology): Let $X$ be a normed $\mathbb K$ -linear space . The weak topology of $X$ is the smallest topology $J_{\omega}$(which is a subset of the normed topology of X) such that $f:(X,J_{\omega})\to\mathbb K$ is continuous for all $f \in X^*.$  Please someone give some hints.  Thank you.
I want to prove that in an infinite dimensional normed space $X$, the weak closure of the unit sphere $S=\{ x\in X : \| x \| = 1 \}$ is the unit ball $B=\{ x\in X : \| x \| \leq 1 \}$.  $\\$  Here is my attempt with what I know:  I know that the weak closure of $S$ is a subset of $B$ because $B$ is norm closed and convex, so it is weakly closed, and $B$ contains $S$.  But I need to show that $B$ is a subset of the weak closure of $S$.  $\\$  for small $\epsilon &gt; 0$, and some $x^*_1,...,x^*_n \in X^*$, I let $U=\{ x : \langle x, x^*_i \rangle &lt; \epsilon , i = 1,...,n \} $  then $U$ is a weak neighbourhood of $0$  What I think I need to show now is that $U$ intersects $S$, but I don't know how.
Ubuntu halts after few seconds of starting and nothing works command, virtual terminal, GUI How do you search for questions filtered by a particular user? I want to do a search on StackOverflow for a few terms, but also by user, since I know which user referenced the item I'm looking for. What should a main game loop do? Generally, what are the core things that one should do in the game loop, and what are some things that one shouldn't do in the game loop? I installed Windows 7, which ate Ubuntu's boot file. When starting up the computer, it now goes straight to Windows, without giving me the option of booting Ubuntu.   How can I get Ubuntu back? Call $P(k): k!&lt;k^k$, for $k\geq 2$ Test it out with 2, and it's true ($2&lt;4$).   Assume that $P(k)$ is true for some $k\geq2$.  Then show that $P(k+1)$ is true.   $P(k+1): (k+1)!&lt;(k+1)^{k+1}$  Ministep: $(k+1)!=k!(k+1)$ Ministep: $(k+1)^{k+1}=(k+1)^{k}(k+1)$    $k!(k+1)&lt;(k+1)^{k}(k+1)$  Can I just pull out that (k+1) and call it a day? That will make $k!&lt;(k+1)^{k}$.   We're already assuming that $k!&lt;k^k$, for some integer $k\geq 2$, and I don't think I need to prove that $k^{k}&lt;(k+1)^{k}$. I mean, I guess I could go prove that as well, but let's assume for the moment that I don't have to. Since we're assuming the original $P(k)$ is true, I should be able to write $k!&lt;k^{k}&lt;(k+1)^{k}$ right? That would prove the $k+1$ case. Is this the correct way to go about things? It seems too easy, but makes sense.  Thanks for confirmation and/or pointing out mistakes! How to resolve a hostname to an IP address in Metro/WinRT? Create taxonomy terms on the fly Does anyone know the name of travel posters from the 1930s? Is there a name for this design style - use of flat colours for dimension and shading - that was so prevalent in UK travel posters of the 1930s? I'm printing a tradeshow flyer and was wondering what type of paper I should print it on. I don't want anything glossy so I'm trying to decide on the following:   100# text weight matte 80# cover 80# matte   What is the difference between the types above? And how can I make a decision? Is there a JavaScript equivalent of Java's class.getName()? What the #$@&%*! is that called? Is there a name for the use of symbols in place of curse words, for example #$@&amp;%*!? For example, is it possible to drink a flying potion and cast greater invisibility spell on myself (both spells would require concentration), so I can fly around invisible? What is the purpose of the thin wire at the top of power transmission lines? I know a power transmission line carries usually two sets of 3 phase conductors. But I have noticed a relatively thin wire at the center-top of the power transmission line. What is the purpose of this thin wire? The median minimizes the sum of absolute deviations (the $ {\ell}_{1} $ norm) Suppose we have a set $S$ of real numbers. Show that   $$\sum_{s\in S}|s-x| $$   is minimal if $x$ is equal to the median.    This is a sample exam question of one of the exams that I need to take and I don't know how to proceed. I have secured a visa for the US, be it an ESTA or another visa requiring me to go for an interview. My understanding is that obtaining a visa for the US does not guarantee me entry into the country. Rather, :  Entering the United States A visa allows a foreign citizen to travel to a U.S. port-of-entry (generally an airport) and request permission to enter the United States. A visa does not guarantee entry into the United States. The Department of Homeland Security (DHS), U.S. Customs and Border Protection (CBP) officials at the port-of-entry have authority to permit or deny admission to the United States. If you are allowed to enter the United States, the CBP official will provide an admission stamp or a paper Form I-94, Arrival/Departure Record.  What happens once I arrive in one such port-of-entry? Will I have to go through customs checks? Will I have to talk to an immigration officer explaining my reasons for visiting the US? What documentation, if any, should I bring? Is the documentation I brought for the interview enough? What if I did not attend an interview since it was not required (for example if I obtained an ESTA online)? difference between not to verb , to not + verb "Name three things that, if you were told were part of the job, would cause you to not take the position. "   I've just come across this sentence while reading the news.   Question is: Why to not is used instead of not to? This usage in this way imposes different meaning? Does Stack Exchange contain any edit suggestion or bug submission system, or such type of discussion forum to inform SO or its administrators about suggestions and bug reports that relate to SO itself? I want to find a set of integer solutions of Diophantine equation: $ax + by = c$, and apparently $\gcd(a,b)|c$. Then by what formula can I use to find $x$ and $y$ ?   I tried to play around with it: $x = (c - by)/a$, hence $a|(c - by)$.     $a$, $c$ and $b$ are known. So to obtain integer solution for $a$, then $c - by = ak$, and I lost from here, because $y = (c - ak)/b$. I kept repeating this routine and could not find a way to get rid of it? Any hint?  Thanks, Chan How to select first occurrence between two patterns including them How can I select first occurrence between two patterns including them. Preferably using sed or awk.  I have:  text something P1 something content1 content2 something P2 something text something P1 something content3 content4 something P2 something text   I want the first occurrence of the lines between P1 and P2 (including P1 line and P2 line):  something P1 something content1 content2 something P2 something I have a problem when I try to start a Workflow 2013 in SharePoint Server 2013. I believe the error is related with permissions, but I cannot be sure.  Here is the error message in the log:  RequestorId: ef9edb7c-cc3d-5e10-3681-eada907596a4.  Details: System.ApplicationException: HTTP 401 {"Transfer-Encoding":["chunked"],"X-SharePointHealthScore":["0"],"SPRequestGuid":["ef9edb7c-cc3d-5e10-3681-eada907596a4"],"request-id":["ef9edb7c-cc3d-5e10-3681-eada907596a4"],"X-FRAME-OPTIONS":["SAMEORIGIN"],"MicrosoftSharePointTeamServices":["15.0.0.4420"],"X-Content-Type-Options":["nosniff"],"X-MS-InvokeApp":["1; RequireReadOnly"],"Cache-Control":["max-age=0, private"],"Date":["Thu, 31 Jan 2013 16:36:44 GMT"],"Server":["Microsoft-IIS\/8.0"],"WWW-Authenticate":["NTLM"],"X-AspNet-Version":["4.0.30319"],"X-Powered-By":["ASP.NET"]}  The HTTP response content could not be read.  'Error while copying content to a stream.'. at Microsoft.Activities.Hosting.Runtime.Subroutine.SubroutineChild.Execute(CodeActivityContext context) at System.Activities.CodeActivity.InternalExecute(ActivityInstance instance, ActivityExecutor executor, BookmarkManager bookmarkManager) at System.Activities.Runtime.ActivityExecutor.ExecuteActivityWorkItem   The question is: What permissions do I need to configure in SharePoint for the WF publisher and the WF manager service account?  I’ve set a WF Manager service account named “wfservice” (this account is a local admin and has the sysadmin permissions in SQL Server), and I published the workflow and tested it using the account “wfuser”. Both accounts have full control permissions in the web app and are site owners, for testing WF 2013 purposes.  I’ve already done the DNS flush, as mentioned in Andrew Connell’s article, and I’m still getting an Unauthorized exception in the Item Workflow Status.  Anyone knows what’s going on? Which permissions do I need to configure?