I am having problems to prove the following:     Let V be a K-Vectorspace and $dim(V)=\infty$. Prove that the canonical linear transformation $e: V \rightarrow V^{**}$ is not surjective.     I think I get the idea of the proof, since if $(v_{i})_{i \in I}$ is a basis of $V$ and $(v_{i}^{*})_{i \in I}$ is the dual family in $V^{*}$ then it might - or even can't - span the whole space $V^{*}$. But I am neither sure how I can show this, nor the actual proof. Help would be much appreciated. Thanks in advance!
For the finite-dimensional case, we have a canonical isomorphism between $\mathbf{V}$, a vector space with the usual addition and scalar multiplication, and $(\mathbf{V}^*)^*$, the "dual of the dual of $\mathbf{V}$." This canonical isomorphism means that the isomorphism is always the same, independent of additional choices.   We can define a map $I : \mathbf{V} \to (\mathbf{V}^*)^*$ by $$x \mapsto I(x) \in (\mathbf{V}^*)^* \ \text{ where } \ I(x)(f) = f(x) \ \text{for any } \  f \in \mathbf{V}^*$$  My Question: what can go wrong in the infinite-dimensional case? The notes I am studying remark that if $\mathbf{V}$ is finite-dimensional, then $I$ is an isomorphism, but in the infinite-dimensional case we can go wrong? How?
How to find solutions of linear Diophantine ax + by = c? 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 Is the collection of all functions between two sets a set? The search parameter isanswered ignores acceptances What are the limitations on the Shape Water cantrip? I have been having a discussion regarding the Shape Water cantrip from the Elemental Evil Player's Companion. A friend and I are trying to come to an accurate view of how this spell actually functions.   Some example questions:   Must the targeted water be within a 5 foot cube area, or does it need to be of a volume not exceeding 125 cubic feet? If you shape water that is within a 5 foot cube, can the final shape you animate it to extend outside the cube? Can you cause the shaping to affect ice and snow, or only liquid water? Would you be able to carry a globe of water alongside you over distances?   Really, there are two parts:    What can be targeted by the spell? What are the limitations on the 'shape and animate' portion of the spell?   The answer I am looking for, if possible, would use the language of the spell, possibly with precedent from other game text, to clearly define the limitations of this cantrip. I have created some search entries in Google Chrome using 'Edit search engines'.  How can I share some of these entries with my colleagues? How to handle users who don't select answers for their questions? Chat gives me really, really specific time gap notifications I read here at  that holding Shift during boot can bring up the GRUB menu. However I try holding one or the other Shift keys, tapping, tapping then holding, nothing works. Am I missing something? How does one access the GRUB menu during boot time?   No dual boot here, just standard Lubuntu 14.04 install. Partial Sums of Geometric Series Is WD-40 really bad to apply on bikes? I was told so but still see people doing that.   Also I saw WD-40 launching their new products of bike lubricant and degreaser. Did anybody try those? Rendering order in an Entity System Say I use a , and also inside Systems I hold lists of all entities that Systems are required to process.  How do I maintain this list of entities in desired rendering order, i.e. for a dumb 2D RenderingSystem?   I saw , and what they suggest is to do something like Z ordering - what I would probably do is just to store a "layer" int in DrawableComponent and then, inside RenderingSystem, just sort entities by mentioned "layer" whenever the entity list for RenderingSystem changes.  They also say we could just delete and recreate the entity whenever we want it on the top, but it seems too inflexible to me.  How is this problem usually solved? Convert a string to an enum in C# What's the best way to convert a string to an enumeration value in C#? I have an HTML select tag containing the values of an enumeration. When the page is posted, I want to pick up the value (which will be in the form of a string) and convert it to the corresponding enumeration value. In an ideal world, I could do something like this: StatusEnum MyStatus = StatusEnum.Parse(&quot;Active&quot;);  but that isn't a valid code. Okami is now reborn in High Definition for PS3. The graphics are obviously better, sharper and yet preserves the gist of the original game, however I can swear some of the texts have changed as well.  My PS2 console is no longer available to me so I can't test it myself, but does anyone know whether there are any changes to the game (aside from graphics) or not? When I double-click to select text in GNOME Terminal, the selection stops at spaces but continues over hyphens:      Some of my filenames contain uncommon characters such as the  and are not selectable by double-clicking:    Is there a way to make double-click selection continue over these characters as well? How to sum $\sum_{x=0}^{N-1} \cos{\left( \frac{2\pi x}{N} \right)}^{2}$? I've tried solving it, but I'm not familiar with these kind of progressions and I couldn't get to a result. Also, I've found that the result is just $N/2$, but I don't get why would that be the answer, as it doesn't work for $N=1$ or $N=2$. Appreciate your help :) Using Moq to determine if a method is called It is my understanding that I can test that a method call will occur if I call a higher level method, i.e.:  public abstract class SomeClass() {         public void SomeMehod()     {         SomeOtherMethod();     }      internal abstract void SomeOtherMethod(); }   I want to test that if I call SomeMethod() then I expect that SomeOtherMethod() will be called.   Am I right in thinking this sort of test is available in a mocking framework? Is accessing a variable in C# an atomic operation? I've been raised to believe that if multiple threads can access a variable, then all reads from and writes to that variable must be protected by synchronization code, such as a "lock" statement, because the processor might switch to another thread halfway through a write.  However, I was looking through System.Web.Security.Membership using Reflector and found code like this:  public static class Membership {     private static bool s_Initialized = false;     private static object s_lock = new object();     private static MembershipProvider s_Provider;      public static MembershipProvider Provider     {         get         {             Initialize();             return s_Provider;         }     }      private static void Initialize()     {         if (s_Initialized)             return;          lock(s_lock)         {             if (s_Initialized)                 return;              // Perform initialization...             s_Initialized = true;         }     } }   Why is the s_Initialized field read outside of the lock? Couldn't another thread be trying to write to it at the same time? Are reads and writes of variables atomic? Removing an Ubuntu installation I first had windwos 7, then I installed Ubuntu alongside and when my laptop starts I can choose between ubuntu and windows (the ubuntu boot screen) .  Now I have installed another ubuntu inside the windows 7 using virtual box. I want to remove that first ubuntu installation and free up its partiotion. Can I do it without making any problems for the windows and boot screen? How do I set up Ruby on Rails? I am interested in setting up a Ruby on Rails environment on my 10.10 laptop.   Is there anything special I need to keep in mind for installation (i.e. that could be new or unique to 10.10)?   Are there any suggested walkthroughs that I could follow along with?  Thanks in advance. Indenting a whole paragraph How do I indent a whole paragraph at the same distance from my left margin, with the simplest command or environment?