Let $H$ a Hilbert space.   Q1) Does $H$ has an orthogonal/orthonormal basis ? I know that it's true if $H$ has finite dimension, but what happens in infinite dimension ? Because Gramm-Schmidt is not applicable anymore...   Q2) If the answer to my previous question is yes, does it still hold if we suppose $H$ being only an inner space (with infinite dimension) ?
The problem is to prove that every Hilbert space has a orthonormal basis. We are given Zorn's Lemma, which is taken as an axiom of set theory:  Lemma If X is a nonempty  with the property that every  of X has an upper bound in X, then X has a maximal element.   Given a orthonormal set $E$ in a Hilbert space $H$, it is apparently possible to show that $H$ has an orthonormal basis containing $E$.   I tried to reason as follows: Suppose $E$ is a finite set of $n$ elements. Then one can number the elements of $E$ to create a totally ordered set of orthonormal elements. Then the span $&lt;E&gt;$ can be identified with $R^n$, where each element $v = v_1 e_1 + v_2 e_2 + ... + v_n e_n$ is identified with the vector $(v_1, v_2, ..., v_n)$. On $R^n$ we have a total order, namely the "lexigraphical order" $(x_1,x_2,...,x_n) \leq (y_1,y_2,...,y_n)$ if $x_1 &lt; y_1$ or if $x_1 = y_1$ and $x_2 &lt; y_2$ or if $x_1 = y_1$, $x_2 = y_2$ and $x_3 &lt; y_3$ and so on. Hence $E$ is a totally ordered subset of $H$ and $H$ is a partially ordered set. However, this set doesn't seem to have an upper bound. The set $E$ does have an upper bound. If we define a total order on $E$ only, then X is a partially ordered set satisfying the criteria so X has a maximal element?   This is as far as I got, and I am not sure the entire argument is correct. I don't see how what kind of maximal element I am seeking, since the orthonormal basis of a Hilbert space can have countably infinity number of elements.
Where can I get a clean ISO of a specific build of Windows 10? INSERT takes over 5 hours, when inserting more than specific number of rows When inserting less then about 1,350,000 rows to the table it all takes about 2 minutes, however when number of inserted rows is bigger, then the time needed to insert data grows to about 5 hours.  The problem is not connected with query or indexes, because everything has been working fine for a long time and nothing has changed in the structure of query, tables or indexes.   Problem appeared for the first time about 2 weeks ago and it appears repeatedly on days, when number of inserted rows is bigger than +-1,350,000. For example on one day number of inserted rows is 1,200,000 and the process takes 2 minutes, on the other day number of rows is 1,450,000 and it takes 5-6 hours to insert data.  I tried to rebuild indexes, but it has not helped. I use Chrome as my main browser.  I also spend a lot of time on a particular webapp that my company uses.  I'd like to keep those two things separate.  I can run the webapp in a separate window - but I'd like to go further than that.  I'd like to have it be identified by a different task in the Win 7 taskbar; ideally a different icon; and ideally a different title in Process Monitor.  How can I run a second, independent instance of Chrome, with a different title and icon, that won't merge with my main Chrome browser. Why does flag marking as helpful/declined not always correlate with moderator action? Why does the flash freeze a picture? Why does a flash freeze a picture even if I use a shutter speed like 1/10 sec?  I don't understand it, because the light will hit the sensor the whole time, not just when the flash is fired. Why isn't a virus "alive"? $ KC $ spaces imply $ US $ spaces , but vise versa is false. In the $ US $ space , each convergent sequence has unique limit.      In the $ KC $ space , every compact subset is closed.      It easy to show that $ KC $ spaces imply $ US $ spaces.   The example below show that $ US $ spaces do not imply $ KC $ spaces.      Let $ X = [ 0, 1 ] $ with the the usual topology and attach  a new point z whose neighborhood are open dense subsets of $ [ 0,1 ]$. Observe $ [ 0,1 ]$ is a compact non-closed  of $X$ and then $X$ is not a $KC$ space. However no sequence in $ [ 0,1 ]$ converge to $z$ and in particular  all convergent in $X$ have unique limits.   can you give me more details?   Why $X$ is not closed? Why no sequence in $ [ 0,1 ]$ converges to $z$ and in particular all convergent sequences in $X$ have unique limits ? Add warning when changing display name Is casting the same thing as converting? In Jesse Liberty's Learning C# book, he says "Objects of one type can be converted into objects of another type. This is called casting."  If you investigate the IL generated from the code below, you can clearly see that the casted assignment isn't doing the same thing as the converted assignment. In the former, you can see the boxing/unboxing occurring; in the latter you can see a call to a convert method.   I know in the end it may be just a silly  semantic difference--but is casting just another word for converting. I don't mean to be snarky, but I'm not interested in anyone's gut feeling on this--opinions don't count here! Can anyone point to a definitive reference that confirms or denies if casting and converting are the same thing?      object x;     int y;      x = 4;      y = ( int )x;      y = Convert.ToInt32( x );   Thank you  rp  Note added after Matt's comment about explicit/implicit:  I don't think implicit/explicit is the difference. In the code I posted, the change is explicit in both cases. An implicit conversion is what occurs when you assign a short to an int.  Note to Sklivvz:  I wanted confirmation that my suspicion of the looseness of Jesse Liberty's (otherwise usually lucid and clear) language was correct. I thought that Jesse Liberty was being a little loose with his language. I understand that casting is routed in object hierarchy--i.e., you can't cast from an integer to a string but you could cast from custom exception derived from System.Exception to a System.Exception.   It's interesting, though, that when you do try to cast from an int to a string the compiler tells you that it couldn't "convert" the value. Maybe Jesse is more correct than I thought! Please do not close this as a duplicate unless you can remove the “status-completed” tag from the  (because it’s obviously not been completed).  Steps to reproduce:   Have someone post an answer. Downvote the answer. Have the someone edit the answer within the initial 5 minutes of the answer’s existence. Try to undo the downvote more than 10 minutes later.   You cannot undo the downvote. Evaluate this power series Evaluate the sum  $$x+\frac{2}{3}x^3+\frac{2}{3}\cdot\frac{4}{5}x^5+\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}x^7+\dots$$  Totally no idea. I think this series may related to the $\sin x$ series because of those missing even powers. Another way of writing this series:  $$\sum_{k=0}^{\infty}\frac{(2k)!!}{(2k+1)!!}x^{2k+1}.$$ How do I backup a LVM partition to an image for recovery purposes?  I am trying to run dd on /dev/sda2, but it crashes after ca 8 hours, at around 380G   sudo lvmdiskscan   /dev/centos/swap [       3.89 GiB]    /dev/sda1        [     500.00 MiB]    /dev/centos/root [      50.00 GiB]    /dev/sda2        [     465.27 GiB] LVM physical volume   /dev/centos/home [     411.38 GiB]    /dev/sdb1        [     931.51 GiB]    3 disks   2 partitions   0 LVM physical volume whole disks   1 LVM physical volume   Anyway, could running from a stick make any difference? What is a "closed" or "duplicate" question? What's the past participle form of the word exit? Is it exit (irregular, like set)? exited? exitted? On one page I found exited but if that's the case why isn't it exitted (double t) like with the word emit - emitted? Is there a rule when the consonant at the end is doubled and when not? It's so cool that the notification reminds you of the colour of badge you've earned. But I am a little puzzled by the symbols:      Is it feasible to use the symbol from the earned site? At the moment it uses the symbol from the site I happen to be on (as you can see by looking to the left of the dropdown in each case.) There are several distinct usages:   kernel density estimation   kernel trick   kernel smoothing   Please explain what the "kernel" in them means, in plain English, in your own words. Integral of the square of the normal distribution. Does someone know what the integral of the square of the normal distribution with standard deviation sigma over the range $-\infty$ to $+\infty$ is?  Thanks! Continuous injective map is strictly monotonic What are the solutions to $z^4+1=0$? The PS4 is now released in the EU, but due to shipment shortages is not available for sale unless pre-ordered a long time ago. Luckily, it seems as though Dualshock 4 controllers are availble for sale in a number of shops already.  I've long decided to buy the PS4 as well as a second Dualshock 4 controller, so I've been considering buying the "second" controller now in order to replace my old Xbox 360 controller, whose D-Pad is useless and whose RB button no longer responds correctly. But before I buy the Dualshock 4 early, I wanted to know how well my PC will support it.  I already know that , but what exactly does work, and what doesn't?  Can I expect the same key-mapping on the PC as on a PS4?  Basically, can I replace my Xbox 360 controller with a Dualshock 4 and still play games like Batman Arkham Asylum/City/Origins as if I were using the old Xbox 360 controller?