Let $n$ be a given natural number, and let $X$ denote the vector space consisting of the zero polynomial and of all  polynomials of degree at most  $n$, with real or complex numbers as co-efficients, and defined on a given closed interval $[a,b]$ of the real line, with the inner product defined by  $$\langle x, y \rangle \ \colon= \  \int_a^b \ x(t) \ \overline{y(t) } \ \mathrm{d} t   \   \   \ \mbox{ for all } \ x, y \in X.$$  Then is $X$ complete with respect to the norm induced by the above inner product?   I know that the space of all continuous functions is not complete in the above norm.   Let $x_m \colon= \sum_{j=0}^n \alpha_{jm} t^j$, where $t \in [a,b]$ and $m \in \mathbb{N}$,  be a Cauchy sequence in $X$. Then, given $\epsilon &gt; 0$, there is a natural number $N = N(\epsilon)$ such that  $$\Vert x_m - x_k \Vert &lt; \epsilon \ \ \ \mbox{ for all } \ m, k \in \mathbb{N} \ \mbox{ such that } \ m &gt; N \ \mbox{ and } \ k &gt; N.$$ Or,  $$\sqrt{\int_a^b \ \vert x_m(t) - x_k(t) \vert^2 \ \mathrm{d} t } &lt; \epsilon \ \ \ \mbox{ for all } \ m, k \in \mathbb{N} \ \mbox{ such that } \ m &gt; N \ \mbox{ and } \ k &gt; N.$$ That is,  $$\sqrt{\int_a^b \ \left\vert \sum_{j=0}^n \left( \alpha_{jm} - \alpha_{jk} \right) \  t^j \ \right\vert^2  \ \mathrm{d} t } &lt; \epsilon \ \ \ \mbox{ for all } \ m, k \in \mathbb{N} \ \mbox{ such that } \ m &gt; N \ \mbox{ and } \ k &gt; N.$$  What next?   Our aim should be to achieve the "Cauchy-ness" of the sequence $\alpha_{jm}$ of real or complex numbers, for each $j = 0, 1, \ldots, n$.   Am I right? If so, how to achieve this goal?
A Hilbert space is, by definition, a complete inner product space. If $(V,|.|)$ is finite dimensional inner product space of dimension $n$ then it is (topologically) isomorphic to $\mathbb{R}^n$ which is of course complete. My instinct here is to say "and therefore, $V$ is also a Hilbert space". I'm not sure about this step however since completeness depends on the norm and the norm depends on the selected inner product (assuming the induced norm $||x|| = (x | x)^{1/2}$ is used). So, must we place some condition on the inner product or is the topological isomorphism between $\mathbb{R}^n$ enough to guarantee that V is Hilbert?
I have been trying to install ubuntu (I tried 12.04LTS, 12.10, 12.10-secure-remix) using the wubi installer on Windows 8 and all the three times ran into the same problem showing some cannot find /ubuntu/winboot/wubildr when I tried to start ubuntu.  I tried to change the boot settings from UEFI to Legacy and then my laptop restarted and showed that "No operating System Found".  I then tried to install Ubuntu using a cd and after a lot of effort and using boot repair,etc. I landed into a new trouble. When I select UEFI on, the laptop loads Windows and windows too does not detect ubuntu. When I select Legacy mode, it starts ubuntu, shows me an option to start windows 8 but gives an error showing windows failed to load.  So in legacy mode, i am able to use only ubuntu and in UEFI mode, I am able to use only Win8. Please explain any process of starting windows in legacy mode (which I tried without any success) or starting ubuntu in UEFI mode. Why did the TIE fighter in The Force Awakens have life support? Please help me to find elements $a,b$ in a ring $R$ such that $a\mid b$ and $b\mid a$, but there does not exist any unit $u$ in $R$ such that $a=ub$. Why does it says "welcome back -user- click here to 'refresh page'" After logging into one Stack Exchange site, when I visit another SE site, it says to me, welcome back - user - click here to refresh - web page.   Isn't it possible to refresh the page using code &amp; without the user's interaction? Inform moderators migrating a question if the migration would have been normally blocked HTML Element Array, name="something[]" or name="something"? Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements from $k[X_1,X_2,X_3,X_4]$? Can $(X_1,X_2) \cap (X_3,X_4)$ be generated with two elements in the ring $R=k[X_1,X_2,X_3,X_4]$? Can it be generated with three elements? (Here $k$ is a field.)    Thanks for any help. Novel about a dragon lord who has to wear gloves at all times Surface of genus $g$ does not retract to circle (Hatcher exercise) I'm trying exercise 9 on page 53 in  but I need some help with it. The exercise is:  In the surface $M_g$ of genus $g$, let $C$ be a circle that separates $M_g$ into two compact subsurfaces $M_h^\prime $ and $M_k^\prime$ obtained from the closed surfaces $M_h$ and $M_k$ by deleting an open disk from each. Show that $M_h^\prime$ does not retract onto its boundary circle $C$, and hence $M_g$ does not retract onto $C$. [Hint: abelianize $\pi_1$.] But show that $M_g$ does retract onto the nonseparating circle $C^\prime$ in the figure.    My first question is: assume there wasn't the hint, how would I think of abelianising? What does it mean exactly?  I thought I could do this by contradiction: if it retracts the induced map $i_\ast : \pi_1(C) \rightarrow \pi_1(M_h^\prime) $ is injective. I know $\pi_1(C) \cong \mathbb{Z}$, then I computed $\pi_1(M_h^\prime) \cong \mathbb{Z}$ and then I think I'm stuck. Right? Do you agree with $\pi_1(M_h^\prime) \cong \mathbb{Z}$ and being stuck after that?  What do I need to know to make progress? Many thanks for your help! Why does this excerpt say that unbiased estimation of standard deviation usually isn't relevant? How to show bibliography alphabetically with biblatex? Determinant of a block lower triangular matrix I'm trying to prove the following: Let $A$ be a  $k\times k$ matrix, let $D$ have size $n\times n$, and $C$ have size $n\times k$. Then,  $$\det\left(\begin{array}{cc} A&amp;0\\ C&amp;D \end{array}\right) = \det(A)\det(D).$$  Can I just say that $AD - 0C = AD$, and I'm done? OpenLayers and OSGB 1936 (CRS 27700) I've been working with loads of shapefiles projected according to OSGB 1936 (CRS 27700). The project CRS is set to EPSG:27700 and "On the fly reprojection" to EPSG:27700 is set (I keep this as a default as I work with OSGB 1936 95% of my time).  I decided to download OSM layers from the OpenLayers plugin. When I load this, the EPSG in the bottom right of the screen goes from EPSG:27700 to EPSG:3857   Question 1: should this be happening anyway as it would suggest that the whole project has been re-projected to EPSG:3857?   As soon as the OSM layer is loaded it's way out of alignment but a quick pan seems to bring it altogether quiet nicely.   What I have noted is that (Question 2), if I try to re-specify the CRS to 27700, then the shapefiles "move" by a few tens of meters with respect to the OSM. Does this suggest that QGIS is doing a second re-projection (from what to what?)?   Also (Question 3), does it mean that if I import another shapefile that would be originally a dataset digitised with EPSG:27700 it would be re-projected to EPSG:3857 because the CRS in the bottom right of the screen was set to EPSG:3857 after the OSM import. This is all a bit confusing and logic would suggest that the import of a non-ESPG:27700 (project default CRS) should not go and change the EPSG setting in the bottom right of the screen. Anita Graser's book says on page 34 "Your project CRS will automatically be changed to Pseudo Mercator when you load a background map using the OpenLayers plugin option".   Does that mean there is no way of keeping ESPG:27700 active in the project when importing basemaps from OpenLayers? If so, what alternatives are there?  What I am trying to bottom out is to understand the correct way of importing basemaps (usually Google, Bing or OSM through the OpenLayers plugin) into my projects (always EPSG:27700) without losing alignment and getting layers to correctly re-project. I'm trying to support greater use of QGIS in my company but don't want to lose credibility by promoting the use of a tool that is misaligning datasets. Don't get me wrong. QGIS is a great tool and I am very fond of it but importing layers with different CRS's is quite common these days and getting this right from the start is half the job done.  I did quite an extensive Google search before posting and didn't come up with a recent, clear cut answer to my questions above. There is a bit of content out there on problems with older versions of QGIS, PROJ4, etc. but nothing recent for QGIS 2.0 so worth an update. Is it possible to mark or segregate bad blocks on HDD? Hello I want to know if is it possible to mark bad blocks on HDD for OS to not use them. I don't want to save data written on this blocks, just want to get rid of them.  I found some article where author describe how to partition disk to avoid using bad blocks of HDD.  I count how many MB from the start are OK. And also how many MB from end are OK. It looks like I will lost about 50GB of storage space but it is quite fine for me when the disk will run one year more.  First I split disk to 2 partitions:   From beginning to around 190 GB From 190GB to the end   Then I made surface scan of 2nd partition and everything was OK. I was really happy cause i though that finally I will succeed.  Than I wanted to copy all data from 1st partition to 2nd BUT in some data a problem occurs. I/O error. I'd say it is Read problem but when I open file on which copying failed it runs OK, for example some pdf files didn't copied but I opened them.  What can be problem can it be a Write problem? But why there is a problem when surface scan on whole 2nd partition was fine?  What can I do(except buying new HDD)? Can I somehow do this partitioning and use just partition with good sectors? Will it work? How are data stored on the disk? Can I determine where the data will be written? What's the difference between "either" and "neither"? Run VirtualBox in background, without a window? I was wondering if it's possible to run an Ubuntu virtual machine using VirtualBox in background, I mean, without any window open.  The idea is to connect via SSH to the Linux host, that would be running in background.  Has anyone ever done something like this? Is it possible? I am interested in comparing the computational cost of grid search vs. random search algorithm for hyperparameter tuning. Is random search computationally less expensive then grid search? Is there an article/paper that talks about this topic? Thank you, When Frodo put on the One Ring, both the Nazgul and Sauron himself found Frodo. However when Bilbo had the ring and possibly used it many times, Sauron sought Gollum to get a hint on where to look for the ring and send the Nazgul after it.   Sauron sought Gollum before Bilbo wore the ring at his birthday party, slightly after that time, he was very effective at locating Frodo when Frodo put on the ring. So I guess it's not a matter of Sauron gaining any extra powers with time.  Also Bilbo seemed to use the ring without it bothering him at any point, unlike Frodo who seemed to be very troubled when using the ring. How is that possible? How to correctly add a path to PATH? I'm wondering where a new path has to be added to the PATH environment variable. I know this can be accomplished by editing .bashrc (for example), but it's not clear how to do this.  This way:  export PATH=~/opt/bin:$PATH   or this?    export PATH=$PATH:~/opt/bin A word similar to "Donate" but more freely provided to