Is there a simple proof showing that $\dim V^*=\dim V$? Where $V^*$ denotes the dual space of a finite dimensional space $V$?
So, we know that there is no "natural" isomorphism between a finite dimensional vector space $X$ and its dual $X^{\vee}$. However, given a basis for $X$ $(e_1, \dots, e_n)$ we can always construct a (dual) basis $(e^1, \dots, e^n)$ for $X^{\vee}$ that satisfies $e^i(e_j) = \delta^i_j$. It follows that $V \approx V^{\vee}$ simply because they have the same finite dimension $n$. Now, it seems to me that we can construct an isomorphism by assigning $$ e_i \mapsto e^i $$ and extending by linearity. That is, for an arbitrary vector $v \in X$ given by $v = \sum v^ie_i$ we can specify a linear map   $$\phi:V \rightarrow V^{\vee}$$  by  $$ \phi(\sum v^ie_i) := \sum v^ie^i $$ It is a quick exercise to check that $\phi$ is an isomorphism. Now, although this construction seems correct, there is something that doesn't quite sit right with me about it. For one thing, there is no way to really make sense of the summation convention as the right side of the function will always contain two superscripts thus requiring an explicit summation sign. The other thing that feels rather off is actually taking a lower index and moving it to an upper index, i.e., $e_i \mapsto e^i$ (which, of course, is the reason that the summation convention doesn't work).  So, with this background my questions are   Is there any sense in which the above isomorphism is "favored" or "canonical"? Is there another way to construct an isomorphism between a vector space and its dual, that would "conserve indexes", for lack of a better way to state it. My guess here is that the answer is no but becomes possible if one assumes $V$ has an inner product.
Does quantum spin contribute to the angular momentum of a black hole? How could I adjust the size and placement of chapter heading in my custom class? I am using report class as my base. When I used titlesec package inside my custom class, it threw error     ! Package titlesec Error: Not allowed in `easy'   settings. Will two-letter font style commands (\bf , \it , …) ever be resurrected in LaTeX? I understand that \bf and \it are now obsolete in LaTeX and that \textbf and \textit are proper, as they produce more sophisticated (in particular, cumulative and properly kerned) changes to font style. I have read the  and I understand the rules and their reasons, as well as the several other commands that are affected.  However, I find it convenient to use LaTeX for notetaking during lectures, and in that rushed environment, shortening a command by any number of keystrokes helps keep me from falling behind. \bf and its two-letter kin are still very useful to me for that reason, and once a presentation is finished I can go through and replace all appearances of \bf et al. with \textbf et al.  My question is this: is there a plan eventually to replace the short font style commands like \bf with the implementations of \textbf etc. some day, or should I expect \bf always to remain in existence but obsolete, for reasons of backward compatibility with original TeX? Original TeX has been greatly improved on in countless ways, but in the heat of transcription I sometimes miss its conciseness. When do you use the "this" keyword? What's the best way to animate an illustration for the web? I have a couple of illustrations done in Illustrator and I am planing to animate it for a website that I am working on, I've heard of  toolkit with flash, but is it the best way to go or is there another "better" way of doing it?  Here's an example of the kind of animations I am aiming for: Add space between headings and body in the home of the iPad app It's a pain in the ass for me to quit the app, draw a red circle in another app and return to the app, so...    If you notice the three questions in the bottom, you will notice that the emboldened headings (New Answer, Comment) are stick to the next character. Imperfect.    App Version: 1.4.0.249 Device: iPad 2 (GSM) OS Version: Version 9.1 (Build 13B143) Two of my friends are getting married. They're both huge Doctor Who geeks, and actually met through a DW RPG. Their wedding invitation contained a very subtle DW reference (you wouldn't have been able to tell unless you were very familiar with the show yourself.)   Doctor Who's been running long enough that there are quite a few nice quotes for basically every occasion in life. I was hoping someone would know a nice wedding-related one, or maybe one about love, that I could include in a wedding card? Find the value of $\sin\frac{2\pi}{7}+\sin\frac{4\pi}{7}+\sin\frac{8\pi}{7}$ Motivation for Napier's Logarithms In the wikipedia article on logarithms, I am clueless about the approach and motivation for the following computations done by Napier (and the mysterious appearance of Euler's number) in this . Please help me understand:       By repeated subtractions Napier   calculated $10^7(1 − 10^{−7})^L$ for L   ranging from 1 to 100. The result for   $L=100$ is approximately $0.99999 = 1  − 10^{-5}$. Napier then calculated the   products of these numbers with $10^7(1  − 10^{−5})^L$ for L from 1 to 50, and   did similarly with $0.9995 ≈ (1 −  10^{−5})^{20}$ and $0.99 ≈ 0.995^{20}$.   These computations, which occupied 20   years, allowed him to give, for any   number ''N'' from 5 to 10 million, the   number ''L'' that solves the equation      $$N=10^7 {(1-10^{-7})}^L$$      In modern notation, the relation to   natural logarithms is:      $$L = \log_{(1-10^{-7})} \!\left(  \frac{N}{10^7} \right) \approx 10^7  \log_{ \frac{1}{e}} \!\left(  \frac{N}{10^7} \right) = -10^7 \log_e  \!\left( \frac{N}{10^7} \right)$$      where the very close approximation   corresponds to the observation that      $${(1-10^{-7})}^{10^7} \approx  \frac{1}{e}. $$   I understand the computations, but I don't know what's going on. Eigenvectors of real symmetric matrices are orthogonal What's the term when you ask a question which implies a lie? Model a simple 3D spiral shape? I need to model a simple spiral shape. I imagine the best way to do this is with curves, but I am unsure how to create a perfect spiral shape with curves. What's the best/easiest way to make a spiral tube? If a character who was under the effect of a growth potion (double their height and eight times their weight) and weighed 2,400 lbs. was able to use Dimension Door to teleport 400 feet or more into the air, directly above a Huge-sized monster, and fell on it, how much damage would the falling creature - and the monster - take?  I would assume both would take the 20d6 max for the falling over 200 feet, but is there a estimation on additional damage for the falling creature's size?  According to the splat calculator, at 500 feet you'd be falling at 196 km/hr, and it would expend 1.6 million joules of energy, the equivalent of over 3,000 mid-sized cars hitting an object at 60 km/hr.  A 2,400-lb. barbarian falling 500 feet is like dropping a mid-sized Toyota Corolla off a 50 story building onto a monster - it should do some damage. Powershell Parameters Find an element by id and replace its contents with php Show that $[0, 1)\times[0, 1)$ is homeomorphic to $[0, 1]\times[0, 1)$ but not to $[0, 1]\times[0, 1]$. When I sketch these spaces it this statement makes sense intuitively because $[0, 1]\times[0, 1]$ is closed and $[0, 1)\times[0, 1)$ and $[0, 1]\times[0, 1)$ are each "missing" part of their boundary which would make them closed. I also know, $[0, 1]\times[0, 1]$ is compact and compact spaces are not homeomorphic to non-compact spaces.   I am having trouble saying this explicitly, as is pretty obvious in the clumsiness of the wording above. Summing data from column in the Attribute table I'm working with population data, and I have an issue when it comes to knowing the sum of a population inside the attribute table.  In my attribute table, I'd like to add all the numbers in the red box, i.e. all the numbers together from the column &quot;UN_2015_E&quot;. Is there a way to do so? Distributing distinct apples and pears to three people so that each person receives at least one pear How many ways are there to distribute seven distinct apples and six distinct pears to three distinct people such that each person has at least one pear.   Note: The first part of the problem was the same but the word distinct was replaced with identical apples and pears. That answer is $360$ ways.   What has to be done differently when the word identical is replaced with distinct? "Scaled $L^p$ norm" and geometric mean The $L^p$ norm in $\mathbb{R}^n$ is \begin{align} \|x\|_p = \left(\sum_{j=1}^{n} |x_j|^p\right)^{1/p}. \end{align} Playing around with WolframAlpha, I noticed that, if we define the "scaled" $L^p$ norm in $\mathbb{R}^n$ to be  \begin{align} \| x \|_p = \left(\frac{1}{n}\sum_{j=1}^{n} |x_j|^p\right)^{1/p} \end{align} then \begin{align} \lim_{p \to 0} \|x\|_p &amp;= \left( \prod_{j=1}^{n} |x_j| \right)^{1/n}, \end{align} which is the geometric mean of the coordinates' absolute values. This is interesting maybe because the $L^p$ norm doesn't have a nice limit at zero.  My questions:   How do I prove this? Is this definition of "scaled $L^p$ norm" interesting, or known by another name, or used anywhere? Is there any interesting reason to define the $L^0$ norm as the geometric mean, as above? Further reading?   Thanks! We have been asked to prove the above statement through the two sub-proofs:   Firstly we must prove that $e^x$ is increasing and then prove that $\log(x)$ is also an increasing function.   After we are asked to prove that using the delta-epsilion definition of a limit that $\lim_{x \to 0} e^x = 1$.   I have done all of this but still don't see where the instructor is going with this.