Let $W$ be a subspace of a vector space $V$ over a field $F$. Let $i : W \to V $ denote the inclusion map.  Show that $ \pi: V^*\to W^*$given by $\pi(f) = f\circ i$ is a surjective linear map, with kernel equal to $W^0$. Hence show that $W^* = V^*/W^o$.
I wanted to show that $X' / U^\perp \cong U'$, for $U$ being a closed subspace of the Banach space $X$.  Therefore I looked at $l: X' / U^\perp \cong U' , x' + U^\perp=[x']  \mapsto x'|_U$.  It is clear that this map is onto, as we just take a $u' \in U'$. Then I can use Hahn-Banach to get an appropriate $x'$ such that $x'|_U = u'$ and in particular $||x'||=||u'||$.  Now I need to show that is isometric: since we always have $\|[x']\|\le \|x'\|$ and by Hahn-Banach we have $\|x'\|=\|x'_{|U}\|$, this shows $\|[x′]\|\le \|x'_{|D}\|$ and it should be always true that $\forall y' \in U^\perp: \|x'-y'\| \ge \|x'_{|D}\|$, hence $\|[x']\|\ge \|x'_{|D}\|$. Is this reasoning correct? The isometric property follows.
Inverse images of images, images of inverse images of a set of vectors What can be said about$\ f^{-1}(f(S))$ and$\ S$, where$\ f(S):=\{f(x):x \in S\}$?   I feel as if$\ f^{-1}(f(S))$ will be smaller than or equal in size (number of elements) to$\ S$.  For example, if$\ f(x)=x$, then$\ f(\{-1, 0, 1, 2\})=\{-1, 0, 1, 2\}$and$\ f^{-1}(f\{-1, 0, 1, 2\})=\{-1, 0, 1, 2\}$ (same size).  But if$\ f(x)=x^2$, then$\ f(\{-1, 0, 1, 2\})=\{0, 1, 4\}$ and$\ f^{-1}(f\{0, 1, 4\})=\{0, 1, 2\}$, which is a little bit smaller than$\ S$.   What can be said about$\ f(f^{-1}(U))$ and$\ U$, where$\ f^{-1}(U):=\{x \in X:f(x) \in U\}$?   The same size as each other? Always?? proving combinatorics identity - $\sum_{k=0}^m{n-k \choose m-k}={n+1 \choose m}$ How to disable a USB port in Windows (XP) and make it available only for charging? What is the best way to filter a Java Collection? I want to filter a java.util.Collection based on a predicate. Word meaning a job someone is installed in where they don’t do anything With OpenDNS now needing one to be on the pro package to have filtering turned on, being on a tight budget, we are in need of free DNS filtering.  After reading , I collected SquidGuard blacklists and concatenated domains in the categories i needed into one big list. I then fed it to  generating a file containing about 2 million zones.  named-checkconf parses it just fine, but reloading named takes almost 10 minutes, then it wont resolve anything, and the server slows down to a crawl. Removing the include results in named working properly again.  Whats the proper way to set up BIND9 to work as a DNS firewall efficiently? If it's possible to also block any domain that resolves to specific IP addresses that would solve the whole problem.   I'm running bind as non-authoritative, just a caching server. Finite Series - reciprocals of sines Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be $\dfrac{\cos1^{\circ}}{\sin^21^{\circ}}$ but I do not know how. I have tried reducing it using sum to product formulae and found out the actual value and it agrees well. Haven't been successful in telescoping it. Xcode - How to fix 'NSUnknownKeyException', reason: … this class is not key value coding-compliant for the key X" error? Dr. StrangeNumbers or: How I Learned to Stop Worrying and Love Floating Point Arithmetic Spartan 6 DCM unstable clock output class C {   T a; public:   C(T a): a(a) {;} };   Is it legal? Evaluate definite interval of normal distribution I know that an easy to handle formula for the CDF of a normal distribution is somewhat missing, due to the complicated error function in it.  However, I wonder if there is a a nice formula for $N(c_{-} \leq x &lt; c_{+}| \mu, \sigma^2)$. Or what the "state of the art" approximation for this problem might be. Is there a design pattern for dropdown lists in iOS? Time-ordered operator in Srednicki , "Note that the operators are in time order...we can insert $T$ without changing anything". This I agree with. But then on the next paragraph he states "The time order operator $T$, moves all...where they annihilate...". How correct is this paragraph? Is equation (5.14) always valid? Is equation (5.14) the reason why the time-ordering symbol $T$ appears all over the place in QFT?  The equations I'm referring to are (braket package available?):   $$\langle f|i\rangle = \langle0|~a_{1'}(+\infty)a_{2'}(+\infty)a_{1}^\dagger(-\infty)a_{2}^\dagger(-\infty)~|0\rangle \tag{5.13} $$  and since time goes from right to left inside the braket we put the $T$-symbol  $$\langle f|i\rangle = \langle 0|T ~a_{1'}(+\infty)a_{2'}(+\infty)a_{1}^\dagger(-\infty)a_{2}^\dagger(-\infty)~|0\rangle \tag{5.14} $$   without changing anything.   After this, the $T$-symbol appears everywhere! Are Lorentz transformations linear transformations? My textbook says that Lorentz transformations are linear transformations and present them as matrices. Lorentz transformations relate different coordinate systems with each other. It seems that coordinate systems are linear spaces, but coordinates are just labels for spacetime points, with no other structures attached. So what is the linearity all about? How to populate UIList with all material slot in scene? 2.8 SD card partitioning? why? how? Beamer background image centered I'm trying to use an image as a background for Beamer presentation according to  post.  However, I do not want the image spanning the entire background. Thus, I'm doing something like:  \usebackgroundtemplate{% \centering \hspace{4.5cm} \includegraphics[width=1.5in]{image.jpeg} }   The \hspace command works, but \vspace does not work. Basically, I'm trying to get the image to be at the center of the page. Any ideas on how this can be done? How to add Lat/long columns to a point layer which is not in WGS84? I need to extract the Lat/long coordinates of the centroids of districts. My QGIS project SCR and the different layers have all been set on "WGS 84 / UTM zone 18N".   I've created the centroids using the usual tool. I could then extracted the coordinates using the "Vector > Geometry Tools > Export/Add Geometry Columns", or saving the layer in CSV typing "GEOMETRY=AS_XY" in the "Layers" part of the "OGR creation options".  Nevertheless, both methods output Cartesian coordinates, whereas I need Lat/long coordinates. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function. Prove that if a function $f$ has a jump at an interior point of the interval $[a,b]$ then it cannot be the derivative of any function.  I know that for $f$ is differentiable in $(a,b)$ and that it has one-sided derivative $f_+' (a)≠f_-' (b)$ at the endpoints. If $C$ is a real number between $f_+' (a)$  and $f_-' (b)$, then there exists $c∈(a,b)$ such that $f' (c)=C $. How can I use this to prove the above?