Prove that if M and N are closed linear spaces and $M \perp N$, then $M \oplus N$ is a closed linear space.  I'm having trouble starting this one. Do I need to consider cauchy sequences in each M and N?
I'm working on the following problem set.  Let $\mathcal{H}$ be a Hilbert space and $A$ and $B$ orthogonal subspaces of $\mathcal{H}$.  Prove or disprove:  1) $A \oplus B$ is closed, then $A$ and $B$ are closed.  2) $A$ and $B$ are closed, then $A \oplus B$ is closed.  I could prove 1) and if my proof is correct, it even holds if $\mathcal{H}$ is just an inner product space.  Unfortunately, I can't manage to prove 2). Since $\mathcal{H}$ is by assumption a Hilbert space and I didn't use that fact to prove 1), I should probably use it here.  It means that $A$ and $B$ are also complete. Given some convergent sequence in $A \oplus B$, I want to show that the limit is also in  $A \oplus B$. Here I'm stuck. I want to use the completeness of $A$ and $B$, but I don't see how to obtain suitable Cauchy sequences.  Can anyone drop me a hint? Or is my approach all wrong?
How can I enable access to USB devices within VirtualBox guests? I cannot see any USB devices within my VirtualBox guest VMs from my host. How do I enable access for my guest VMs?  &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Show that $K[x,xy,xy^2,\dots]$ is not Noetherian Here is the problem I am stuck on: Fix a field $K$ and consider the subring $A \leq K[x,y]$ generated by $K \cup \{x,xy,\dots,\}$. Show that $A$ is not Noetherian.  I figure that taking ideals $I_n = (x,xy,\dots,xy^n)$ should give an infinite strictly ascending chain, which would establish that $A$ is not Noetherian, but I cannot figure out how to show that each $I_n$ is a proper subset of $I_{n+1}$. Any help here? Proving that a Binary Tree of $n$ nodes has a height of at least $\log(n)$. I installed clang-format on Ubuntu 18.04 LTS with sudo apt install clang-format and the installation was successful. I can execute clang-format-6.0 now, but when I use ClangFormat in vim, it tells me that "clang-format is not found". How can I fix the problem? MLE for Poisson-binomial distribution I am looking for the maximum likelihood estimator (MLE) for the .  I understand the derivation of the MLE for a  and a , but I am unable to derive the MLE equation for the Poisson-binomial distribution. Prove that $\sum_{r=1}^nr(r+1)=\frac{n(n+1)(n+2)}{3}$ using induction $$\sum_{r=1}^nr(r+1)=\frac{n(n+1)(n+2)}{3}$$  could you help me with how exactly I work this out? Disabling middle mouse button Rolle's Theorem and the Mean Value Theorem Let $f\colon \mathbb{R} \to \mathbb{R}$ have derivatives of all orders. Suppose that $a &lt; b$ and that $f(a)=f(b)=f'(a)=f'(b) =0$. Prove that $f'''(c) = 0$ for some $c$ in $(a,b)$. How do I transfer my Destiny account from Xbox 360 to Xbox One? Number of possible permutations of n1 1's, n2 2's, n3 3's, n4 4's such that no two adjacent elements are same? If a character is drunk, what are the penalties associated to it? Are there any benefits to it? I do remember a special prestige class for monks about it, but I don't remember anything specific in the core rulebook. What is "step size" in LASSO regression? How can I determine if a used game is pirated? Terminal returns not found for most commands Mac OSX Gravitational slingshot of light using a black hole/massive object Wikipedia has this page on  using planets. In some cases this effect was used to accelerate the spacecraft to a higher velocity. This diagram shows this in a very oversimplified manner.   That got me thinking that if light is affected by gravity, and if it slingshots around the black hole/massive object, can't it gain a higher speed than $c$? What limitations are stopping it from doing this?  Forgive me if the answer to this question is pretty straight-forward or staring-you-in-the-face kind. I haven't fully understood the mechanics of the gravitational slingshot fully yet, but I couldn't wait to ask this. Note: If it is possible (I highly doubt that!), could you provide an explanation using Newtonian mechanics, I'm not very familiar with general relativity because I'm in high school. Creating a selector from a method name with parameters I have a code sample that gets a SEL from the current object,   SEL callback = @selector(mymethod:parameter2);   And I have a method like    -(void)mymethod:(id)v1 parameter2;(NSString*)v2 { }   Now I need to move mymethod to another object, say myDelegate.  I have tried:  SEL callback = @selector(myDelegate, mymethod:parameter2);   but it won't compile. Ethernet: how important is the magnetics topology? I have some Ethernet issues with a set of boards (6/10 are bad). They're possibly just assembly house mistakes... but I'm very concerned about my magnetics jack.  I could have sworn I saw a example schematic with my jack's topology used with my PHY before, but I can't find it. The PHY's datasheet and suggested magnetics jacks are all identical in topology, and very different than what I have.  I'm using a SMSC  PHY, with a Bel  magnetics jack.  The transformer ratios on my jack vs. a suggested jack for this PHY like the  are the same (1:1).  The inductance is the same, 350uH. However, the choke on mine is on the circuit side, not network side. Also, while the TLA has four 75 ohm resistors from taps tied to a single 1nF/2kV cap, mine has an extra set of 1nF caps between the taps and the 75 ohm resistors.  The main difference is that my jack is a POE jack, and I thought I found an example that used one exactly like it. So, the topology is obviously different just because of that feature.  At the moment, I have 4 good boards, 1 'sometimes' board, and 5 boards that just blink a LED at 11.68Hz representing some error that no datasheet explains.  How important is this? Of course there's no other jack with that footprint.  EDIT  I added some detail on the magnetics jacks above. I was just recommended the SI-60152-F, so I'll compare to that one. Of course like all the other recommendations... it also has no stock at anyone on FindChips.  LATE EDIT  It turns out my problem was the crystal layout that fed the Ethernet PLL.  On the respin, I went with a more standard non-POE jack. Why do "df" and "du" commands show different disk usage? I cannot access ubuntu system on my pc, the error message: "the system is running in low graphics mode" and I tried some commands I searched from internet.   I found out a problems seems there is no available disk space. I used "df" and "du" commands to check, the results are as follow:  du -j --max-depth=1 23G  ./home 3.3G ./usr  ......  28G   and  df -Th  filesystem   Type   size    used   available   use%    mounted on  /dev/sda5    ext4   68G     68G    0           100%    /   how would I clean up the system for more disk space? How to replace an old version of Ubuntu with a newer one from USB/DVD? Best practices for making a WordPress site "movable"? I'm planning to buy a domain and install a WordPress site on it at a web-hosting company.  But first I'll need to iron-out some bugs...  I got a working LAMP-stack at my home-computer, so I'm thinking about first installing my site locally - off the Internet.  I'll be mostly tweaking and making "test-posts" to check that everything works, but I'll probably also make some posts I'd like to keep when I move the site and take it online.  When I'm done chasing bugs; I guess I'll just package the public_html|www directory-tree and upload it to the web-hosting company, and also export the database from my home-computer to the web-hosting.    What I'm wondering if there are some "best practices", some do-s and don't-s, some major pits to take note of?    For example, how can ensure the smooth transition of domain-name in posts and links as such, from localhost to whatever-i-end-up-buying.com (I know from previous attempt of running WordPress at my home computer, that this didn't always work quite well...).