Let $(X,\|\|)$ be a normed space, and  $(X^*,\|\|_*)$ and $(X^{**},\|\|_{**})$ be it's topological dual and topological bidual respectively. Now consider a norm $\rho$ in $X^*$which is equivalent to $\|\|_*.$ Does there exists a norm $\mu$ in $X$ such that $\mu_*= \rho$? (Of course, $\mu_*$ denotes the dual norm of $\mu.$)  I know that if $X$ is reflexive, such a norm indeed exists, and it is just $$\mu(x)=\sup_{\rho(x^*)\leq 1} |x^*(x)|.$$   The question is: can we remove reflexivity? I think that for a proof we could use Goldstine's Lemma. A proof or counterexample would be nice.
I am wondering the following question: Let $X$ be a separable Banach space which is linearly isomorphic to a dual Banach space $Y^*$. Is there a Banach space $Z$ such that $X$ is lineraly isometric to the dual of $Z$: $X=Z^*$.  I think that the answer is no, but I do not have a counterexample. Since $L_1$ is not isometric to any dual Banach space, maybe one can find a dual Banach space which is isomorphic to $L_1$...  To finish, do that change anythink if I suppose $X$ to be almost linearly isometric to $Y^*$ ? By almost linearly isometric I mean that for every $\varepsilon &gt;0$ there exist $T$: $X \to Y$ a linear isomorphism satisfying $\|T\|  \|T^{-1}\| \leq 1+\varepsilon$.
What command should I use to rotate wood to S,N,W,E,D and U?  I read the entire world edit guide and tried some things but all the results are not satisfactory. On the existence of a continuous bijection $f\colon [0,1]\to [0,1]\times [0,1]$ Let $f$ be a continuous function on $[0,1]$ such that $f([0,1])=[0,1]\times[0,1].$ Then show that $f$ is not one-one.  Hints will be appreciated. I installed kali linux OS on my laptop.  The installation went well but when I booted in kali linux the screen was backwards. I signed in as root and the screen flipped back but the mouse is reverse (x &amp; y axis).  When I move my mouse left it moves right when I move it down it moves up. If I want to press a button at the upper right I need to press it at the down left.  The screen is acting backwordes (x &amp; y axis)but I see the graphics straight and the mouse is acting backwordes.  I installed kali linux again with a fresh installation. I have the same problem as before.  I installed kali linux again with a fresh installation. I have the same problem as before.   I searched the web And I don't find information on a problem like this Given that $W$ is a subspace of $V$, prove: $\dim W=\dim V \implies W=V$ Suppose $W$ is a subspace of the finite-dimensional vector space $V$ and $\dim W = \dim V=n$  Let $\{\overrightarrow w_1,\ldots,\overrightarrow w_n\}$ be a basis for $W$ and $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ be a basis for $V$.  The set $\{\overrightarrow w_1,\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ still spans $W$. Since $W$ is a subspace of $V$, $W\subset V$ and all the $w_i$'s can be written in terms of $\{ \overrightarrow v_{1},\ldots,\overrightarrow v_n\}$.  $\therefore$ We can remove one by one all the $w_i$'s from the set $\{\overrightarrow w_{1},\ldots,\overrightarrow w_n,\overrightarrow v_1,\ldots,\overrightarrow v_n\}$ while still having it span $W$, which leaves us with the set $\{\overrightarrow v_1,\ldots,\overrightarrow v_n\}$. We know it to be linearly independent since it is a basis for $V$. Therefore the basis of $V$ is also a basis of $W$. From there on, it is trivial to prove double inclusion $W\subset V$ and $V\subset W$ which gives us $W=V$.  Is my approach correct? I accidentally rsynced .thunderbird to user@host instead of to user@host:/home/user.  Where is .thunderbird now?  Is there a good way to search for the folder? Install python packages for specific python version I understand that in Ubuntu 14.04, you have access to both python 2.7.6 and python 3.4.0. When getting python packages through the command line, how do I make sure that I am installing packages meant for python 2.7.6 and not for python 3.4.0? Let $H$ and $K$ be subgroups of a group $G$, and let $HK=\{hk: h \in H, k \in K\}$, $KH=\{kh: k \in K, h \in H\}$. How can we prove that $HK$ is a subgroup iff $HK=KH$? Browsing the internet from the command line In how many ways can you choose $k$ numbers out of $\{1,2,3,\dots,n\}$ so none of them is consecutive? How to show that $\det(AB) =\det(A) \det(B)$? Given two square matrices $A$ and $B$, how do you show that $$\det(AB) = \det(A)\det(B)$$ where $\det(\cdot)$ is the determinant of the matrix? Basically, I would like to make it so that the lighter a pixel is (the more white it has) the more transparent it becomes. Black pixels would be opaque, white pixels would be transparent.  I currently use the slider in blending options on the first tab to get rid of the white and make it transparent but find it works slightly better if I use the magic wand tool with delete.  I figure that there is some blending option out there but there are so many and I'm not sure which to use to achieve my goal.  (Using Photoshop CS6) Is there a program that can make live USBs for any distro, iso, you get the point. I can only find ones that make live USBs for other linux distributions and none for anything else. What is the relationship between size of object with distance? How does the size/length of an object vary with distance?   Is it a logarithmic relationship? exponential? linear?   I plotted a curve of the size/length of an object for different distances from the camera, and the curve looked exponential/logarithmic. I was trying to understand the reasoning behind that. how to prove $G_1$ and $G_2$ aren't isomorphic? Subscribers currently in a Journey What ways could you create a data extension that contains all the subscribers (SubscriberKeys) that are currently “in” a Journey (Journey Builder)?  The Journey I would like to do this for is a complex activity with multiple decision branches resulting in multiple possible journey durations. For example; if the customer didn’t open Email1 after 2 days, then send Email2; else Exit Journey. If the customer didn’t open Email2 after 2 days, then send Email3; else Exit Journey. Wait 2 more days, then End Journey. The minimum journey duration for the above example is 2 days, while the maximum is 6 days.  Note that I don’t want a list of subscribers that “entered” a Journey, but a list of the ones that are “currently in” a Journey and have not yet exited. Answer must account for non-standard Journey exits including Unsubscribes/Contact Deletes, etc. Code Formating in Comments Set the location in iPhone Simulator I used MiKTeX 2.9 on windows. I have downloaded a package I want to install from the internet (manually, not through MiKTeX). The archive (*.lmza) containing the package was saved to a directory on my C:\ drive. Through MikTeX -> Maintenance -> Settings -> Packages -> Change... I can select the directory in which I have saved the archive containing the package. However, if I choose Packages shall be installed from a directory and choose the location of the package, I get this error message: C:\my\package\is\here does not seem to be a local package repository. What does it take for a directory/folder to become a local package repository? I frequently login to a server, then cd into a specific directory. Is it possible to simplify these two commands into one?   ssh bob@foo   cd /home/guest   I'd like to avoid changing anything on 'foo' if possible, as I'll have to clear it with the server administrator. I use bash, but I am open to answers in other shells. How to run a command that requires sudo at login