Let $(e_k)_{k=1}^{\infty}$ denote an orthonormal set in a Hilbert space $H$. If $(c_k)_{k=1}^{\infty}$ is a sequence of positive real numbers such that $\sum c_k^2&lt;\infty$, then the set   $A=\left\{ \sum_{k=1}^{\infty} a_ke_k: |a_k|\leq  c_k\right\}$ is compact in $H$.  Some hint? Thanks in advance.
Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$  According to the book [Dunford and Schwartz, Linear operators part I, page 453] $C$ is compact in the $\ell^2$-norm, but there is no proof. How can I show that $C$ is indeed compact in $\ell^2$ ?
Is the wave function of a particle re-created after a measurement stops? If $a$ and $b$ are relatively prime, prove that $a + 2b$ and $2a+b$ are also relatively prime or have a $\gcd$ of $3$.  I'm very new to number theory so please don't assume I am familiar with some of the terminology. In the proof of the lemma     For every limit ordinal $\alpha$, $\textrm{cf } \alpha$ is a regular cardinal    Jech says that     It is easy to see that if $\alpha$ is not a cardinal, then using a mapping of $|\alpha|$ onto $\alpha$ one can construct a cofinal sequence in $\alpha$ of length $\leq |\alpha|$, and therefore $\textrm{cf } \alpha &lt; \alpha$.    I've spent a while trying to construct this cofinal sequence and was wondering if I was on the right path:   From a bijection $f$ between $\kappa = |\alpha|$ and $\alpha$ take $S = \{\beta \in \kappa : \ f(\gamma) &lt; f(\beta)$ for all $\gamma &lt; \beta\}$ then if $\xi$ is the order type of $S$ and $g$ is the isomorphism from $\xi$ to $S$ the function $f\circ g$ is a cofinal sequence in $\alpha$ and $\xi &lt; \alpha$.   Is this on the right track, or is there an easier way of doing it? Keyboard macro to change default audio device Windows 7, Logitech G110 keyboard  2 audio devices (headphones and speakers)  I am hoping to find a quick way to macro a key to switch to headphones, and a key to switch to speakers.  Anyone have a good program or anything of that nature?  EDIT:  Using AHK to try and get this task working as hoped is sadly not doing the trick.  I have included a screenshot of my audio set, as well as a screenshot of my script as it sits now.    I have also tried to uncheck "show disabled devices" and modified the script to be Down 1, and Up 1 respectively.  It will work for one device, but it refuses to switch to the secondary. How can I calculate the R-squared of a regression with arima errors using R? If I have an arima object like a:  set.seed(100) x1 &lt;- cumsum(runif(100)) x2 &lt;- c(rnorm(25, 20), rep(0, 75)) x3 &lt;- x1 + x2  dummy = c(rep(1, 25), rep(0, 75))  a &lt;- arima(x3, order=c(0, 1, 0), xreg=dummy) print(a)   .  Series: x3  ARIMA(0,1,0)                      Call: arima(x = x3, order = c(0, 1, 0), xreg = dummy)   Coefficients:         dummy       17.7665 s.e.   1.1434  sigma^2 estimated as 1.307:  log likelihood = -153.74 AIC = 311.48   AICc = 311.6   BIC = 316.67   How do calculate the R squared of this regression? What does "restricted \write18 enabled" mean and why does TexLive keep reporting it? What is the command to remove all files but not directories? Let's say I have a directory tree like this:  FOLDER:     file1     file2     file3     Subfolder1:         file1         file2     Subfolder2:         file1         file2   If I used rm -r FOLDER/*, everything in FOLDER would be deleted including sub-directories. How can I delete all files in FOLDER and in its sub-directories without deleting actual directories? Intuitive explanation of the shape of the Maxwell-Boltzmann distribution At higher temperatures (for an ideal gas), the Maxwell-Boltzmann distribution is spread more widely and has a lower maximum. At lower temperatures, the spread is much more narrow and the peak is much higher. Why is this so? I was thinking that this is to maintain constant area under the curves, but the integration of the curve gives total energy—and total energy should not be constant since temperature is increasing.  So why does the distribution curve change thus? I’m looking for an explanation that’s intuitive or that involves math not surpassing first-year university level calculus. I would appreciate it if it could be explained without the Maxwell-Boltzmann distribution equation, since at this level I’m really treating this qualitatively. Using "align" instead of "array" in an array of equations Essentially as the title suggests - in some commutative ring $K$ (with 0,1), if we have 2 distinct proper prime ideals $\mathfrak{p}_1 \neq \mathfrak{p}_2$, is it necessarily the case (or if not, when is it the case?) that $\mathfrak{p}_1 + \mathfrak{p}_2 = K$ ?  I ask because I'm working through some lecture notes which frequently take some $K$ complete with respect to a discrete valuation, a prime ideal in a valuation ring  $\mathfrak{p} \subset \mathcal{O}_K$, a finite extension $L/K$, and then use the fact $\mathfrak{p}\mathcal{O}_L$ factors uniquely as some $\mathcal{P}_1^{a_1} \ldots \mathcal{P}_r^{a_r}$ (where $\mathcal{O}_L$ is the integral closure of $\mathcal{O}_K$ in $L$). Then we say $\mathcal{P}_i + \mathcal{P}_j = \mathcal{O}_L$ $(\forall$ $ i \neq j)$.   However it isn't clear to me, because we've made so many different assumptions along the way, what the reason is that $\mathcal{P}_i + \mathcal{P}_j = \mathcal{O}_L$. Is it because these primes occur in the factorisation of $\mathfrak{p}$, is it because of some property of Dedekind domains or fields, or is it not really possible to say out of context? Many thanks, and if anything needs clarification please ask. Computing determinant without expansion $$\begin{align}\mathrm D &amp;= \left|\begin{matrix} (b+c)^2 &amp; a^2 &amp; a^2 \\ b^2 &amp; (a+c)^2 &amp; b^2 \\ c^2 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)\left|\begin{matrix} b+c - a &amp; a^2 &amp; a^2 \\ b - a -c &amp; (a+c)^2 &amp; b^2 \\ 0 &amp; c^2 &amp; (a+b)^2 \end{matrix}\right| \\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ b - a -c &amp; a+c - b &amp; b^2 \\ 0 &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\\ &amp;= (a+b+c)^2\left|\begin{matrix} b+c - a &amp; 0 &amp; a^2 \\ 0 &amp; a+c - b &amp; b^2 \\ c - a-b &amp; c - a-b &amp; (a+b)^2 \end{matrix}\right|\end{align}$$  Can $\rm D$ be further simplified without expanding ?  I feel it should be because this was competition question. How to calculate the difference between two dates using PHP? Ways to Install and uninstall programs in deb format Abnormal PREEMPTIVE_OS_AUTHENTICATIONOPS Waits Natural Sort in MySQL Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$ I'm not sure how to go about this. Does it involve using MVT?  I got as far as saying $e = \frac{e^x - e^y}{x-y}$. Proving that $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c)$ where $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k<c<y_k$ Question.  Suppose that a function $f$ is defined on an interval $I$, $c$ is a point of $I$, and {$x_k$} and {$y_k$} are any two sequences in $I$ such that $\lim_{k\to \infty}x_k=\lim_{k\to \infty}y_k=c$ with $x_k&lt;c&lt;y_k$ for all $k$. Prove that, if $f'(c)$ exists, then $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c).$   My Attempt.  Let $\epsilon_k=\frac{1}{k}$ be given.  For $\epsilon_1=1,$ there exists $\delta_1&gt;0$ such that $x_1,y_1\in I, x_1,y_1\in N'(c;\delta_1)$ implies $\vert \frac{f(x_1)-f(y_1)}{x_1-y_1} -f'(c) \vert&lt;\epsilon_1.$  For $\epsilon_2=\frac{1}{2},$ there exists $0&lt;\delta_2&lt;$min$\{\vert x_1-c \vert,\vert y_1-c \vert,\epsilon_2\}$ such that $x_2,y_2\in I, x_2,y_2\in N'(c;\delta_2)$ implies $\vert \frac{f(x_2)-f(y_2)}{x_2-y_2} -f'(c) \vert&lt;\epsilon_2.$  Similarly, for $\epsilon_{k}=\frac{1}{k},$ there exists $0&lt;\delta_k&lt;$min$\{\vert x_{k-1}-c \vert,\vert y_{k-1}-c \vert,\epsilon_k\}$ such that $x_k,y_k\in I, x_k,y_k\in N'(c;\delta_k)$ implies $\vert \frac{f(x_k)-f(y_k)}{x_k-y_k} -f'(c) \vert&lt;\epsilon_k.$  Then by taking the limit to infinity, we obtain $\lim_{k\to \infty}\frac{f(x_k)-f(y_k)}{x_k-y_k}=f'(c).$   It really took me a lot of time to finish the proof and I need this to be checked by an expert:) Is the proof okay? Any response would be appreciated! The recipe that I used did not call for cornstarch or corn syrup. The result is that they are slightly wet. Can I do anything to dry them up, such as dusting them with corn flour?  I did dredge them in a mixture of confectioners sugar and corn flour; it helped a little, but I think I will find a recipe that has corn starch in it for next time! Erlang server, Java client - TCP messages get split? Why do shell script comparisons often use x$VAR = xyes?