It's well known that if $X$ is a Banach space, then $X^{**}$ contains a copy of $X$, so we'll informally say $X\subset X^{**}$. It's also well known that in general we don't have $X=X^{**}$. So what if, starting with a Banach space $X$ such that $X\subset X^{**}$ but $X\neq X^{**}$, you considered the sequence $$X\subset X^{**}\subset X^{****}\subset X^{******}\subset\cdots$$ Not sure why this sequence would be of any interest but...is it possible that the inclusions are all strict in this infinite chain?  EDIT: Some are (justifiably) suggesting that this question is a "duplicate" of the question of whether or not $X$ is reflexive iff $X^*$ is reflexive. While, in some sense, one could make that claim based on how the answer to my question almost immediately follows from the truth of that statement, I wouldn't go so far as to claim it's a duplicate question.
How to show that a Banach space $X$ is reflexive if and only if its dual $X'$ is reflexive?
I'm passing through Canada, on my way to somewhere else. I'm staying only a short time.   Is there a website, or some other way of find out out if I need a visa? How to best respond to a request to send a passport scan by unsecured email? It is not uncommon for travel or holiday accommodation companies to ask for a scan of your passport by email. Sometimes local laws require them to keep a record of the passport numbers of all visitors. I'm happy to provide that information but you usually have to send it in an unsecured email, which may have consequences .  Could these companies be more responsible, and provide a public cryptographic key at least?  Are there other steps you should take to protect yourself, if you do send by unsecured email? Why is "high" pronounced "hiy" but is not spelled that way? What counts as an attack? I was looking over the class feature of the 'fiend' patron warlock in 5e. I'm a bit confused on what actions the player needs to take before they can use the 'Hurl Through Hell' ability on an enemy. The PHB states:     Starting at 14th level, when you hit a creature with an attack, you can use this feature to instantly transport the target through the lower planes.   I'm not clear on what an 'attack' means in this context. Does it have to be a physical attack like a sword or crossbow? Do magical attacks like Eldritch Blast that require an attack roll count? What about spells with no attack roll? Cauchy Problem for inviscid Burgers' equation Consider the Cauchy Problem of finding $u(x,t)$ such that    $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}=0,x\in\mathbb{R},t&gt;0$$ $$u(x,0) = u_0(x), x\in\mathbb{R}$$ Which choices of the following functions for   $u_{0}$ yield a $C^{1}$ solution $u(x,t)$ for all   $x\in\mathbb{R},t&gt;0$         $u_{0}(x)=\frac{1}{1+x^{2}}$   $u_{0}(x)=x$   $u_{0}(x)=1+x^{2}$   $u_{0}(x)=1+2x$.      If I use the existence and uniqueness theorem for Cauchy problem i get the corresponding determinant is non zero so all are true according to me. But in answer key only option 2nd and 4th is given. Please help me to solve the problem. Thanks a lot. Why does the weighing balance restore when tilted and released How to create two partitions? How to replace ${} placeholders in a text file? I want to pipe the output of a "template" file into MySQL, the file having variables like ${dbName} interspersed. What is the command line utility to replace these instances and dump the output to standard output? What is inertia of a body? Is "can not" unambiguous? I always try to remember to use "cannot" when applicable as in "I cannot take an umbrella" (because I do not have one).  I also thought that the problem with "I can not take an umbrella" was that it was ambiguous: it could also mean that I could decide not to take an umbrella. Trying to write unambiguously most of the time, I would therefore never use it.  Now I see it claimed that it is in fact not ambiguous, and only carries the second meaning. Usually I would accept such advice at face value and start using "can not" when the second meaning is unambiguously meant, but this comes from a style guide I have other reasons not to respect very much.  Does "can not" have a single unambiguous meaning?  (I have read  but my question is more about current usage. The length of the answer to that question does confirm my prejudice a bit, though) Change edit approval threshold to (# approve − # reject ≥ N) Currently, the threshold for approving or rejecting a suggested edit is , whichever is reached first.  This means that, when  or just  edits get approved, it's often with a vote count of something like 1 reject / 2 approve (or 2 reject / 3 approve on SO).  Typically, these edits manage to get one or two very quick approve votes from, um, , after which it's very hard for the slower and more careful reviewers to accumulate enough votes to catch up before the edit is approved.  Conversely, when a valid edit gets wrongly rejected, this very rarely happens unanimously either.  More often, you'll see something like 3 votes to reject / 2 to approve, but that single last vote, no matter how well or poorly considered, is enough to tip the scales.  So I'd like to propose a simple change to the approval / rejection threshold:        Suggested edits should be approved when S = (# of approve votes &minus; # or reject votes) &ge; N, where N = 3 on SO and N = 2 on other sites.   Similarly, suggested edits should be rejected when S &le; &minus;N.      No, this change is not going to magically fix all the problems with suggested edit review.  But I believe that it would be a simple and conservative improvement over the current behavior:   For obviously good or bad edits that get approved / rejected unanimously, this suggestion would not change the current behavior in any way. For edits that get one mistaken or careless reject / approve vote (as happens pretty often), an extra approve / reject vote would be required to balance it.  This would slightly slow the review process for such edits, but not excessively so.  Further, I would argue that these are precisely the kind of edits that deserve that little bit of extra attention to make sure the eventual decision will be correct. Edits that are truly controversial, and for which the reviewer community is nearly exactly 50&ndash;50 split between voting to approve and reject, might stay in the review queue significantly longer (maybe twice as long on average) than under the current system.  However, this is a trade-off for the increased likelihood that the eventual decision will actually reflect the majority opinion among reviewers, rather than mere random chance.  In any case, even if reviewers are exactly 50&ndash;50 split, the  under the proposed scheme is only N&sup2;, i.e. 4 votes for other sites and 9 votes for SO.  Even a slight bias in the voting will further reduce this.  Besides, the longer the suggestion remains in the queue, the more likely it gets that someone will short-circuit the process by casting a binding vote (mod / post owner) or by choosing to improve the edit.     Ps. If it is felt that there's an issue with , a simple further chance to address that could be to make each reject vote count as, say, &minus;2 rather than just &minus;1 approval votes (and change the rejection threshold to S &le; &minus;2 &times; N to compensate).  This still wouldn't help with bad edits getting unanimously approved (which isn't really something that can be fixed, except maybe by slugster's cattle prod), but it would mean that each reject vote would now effectively cancel out two approve votes, titlting the balance for disputed edit suggestions in favor of rejection and making it easier for more careful reviewers to counter the "robo-approvers". How to add spaces at every X digit in a binary or hexadecimal number? I would like to format binary or hexadecimal numbers with a space separator at each 2 or 4 digit (or any other value like 8 digit or more).  A small picture is better than a long explanation, here is what I would like to do:    As you can see in this mwe, I use the fmtcount package with padzeroes:      \documentclass{article}     \usepackage[utf8]{inputenc}         \usepackage[T1]{fontenc}               \usepackage{fmtcount}     \begin{document}             Binary value : {\padzeroes[16]\binarynum{129}}$_{(2)}$ %             should display 0000 0000 1000 0001$_{(2)}$              Hexadecimal value: {\padzeroes[8]\Hexadecimalnum{65000}}$_{(16)}$ %             should display  00 00 FD E8$_{(2)}$     \end{document}   What I want to do is displaying numbers with various radix (2, 3, 7, 8, 10, 16) given a decimal number and format these numbers correctly with spaces and leading 0 (without having to perform the conversion by myself).  I already found some posts with similar questions, but none of them achieve this (they display number with thousand separators, like the numprint package, every 3 digits; or they do not make it automatic). Here are these posts :         EDIT two other linked questions:       I also prefer to use pdflatex. Find all $n\in\mathbb N^+$ such that the sum of the digits of $5^n$ is equal to $2^n$.   I've solved this, but I think the proof is a bit weird and I wonder if there's a better one.    My proof:  We can see that $n\neq 1$, $n\neq 2$, $n=3$, $n\neq 4$ and $n\neq 5$.  Let $n&gt;5$. Notice that $5^6$ has $5$ digits, while $\frac{2^6}{9}=7\frac{1}{9}$, so $5^6$ has to have at least $8$ digits in order for the sum of the digits to be able to at least reach $2^6$.  Everytime $1$ is added to the exponent of $5$, the amount of digits $5^n$ gets is at most $1$, while $\frac{2^{n+1}}{9}-\frac{2^n}{9}=\frac{2^n}{9}&gt;3$, so the lowest amount of digits it has to have to reach $2^n$ becomes higher by at least $4$.  When $n=6$, the amount of digits $5^n$ has is already lower than the amount it has to have in order to reach $2^n$, and I've just proved that the amount it has to have in order to reach $2^n$ grows faster than the amount of digits $5^n$ has, therefore $5^n$ will never be able to reach $2^n$ when $n&gt;5$.  Hence the only solution is $n=3$. Windows Spotlight Source and Save locations? Prove that $A+I$ is invertible if $A$ is nilpotent Possible Duplicate:          Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix). Is it acceptable to edit (correct) grammar or spelling? Some individuals may be sensitive to edits of their posts on the basis of spelling or grammar. Some editors may be concerned about what to put in as a reason/explanation for the edit. New editors, in general, are likely uncertain of what the accepted "etiquette" is, on these sites, for such corrections. How would I have a  action that may have some effects on the current page but would also change the URL in the browser so if the user hits reload or bookmark, then the new URL is used?    It would also be nice if the back button would reload the original URL.  I am trying to record JavaScript state in the URL. How can we show that the following sequence converges? $(a_n)$ is a bounded sequence with the following condition  $a_{n+1}\geq a_n-\frac{1}{2^n}$   The sequence converges, but how do we show it? Can I write the output the of time difference command to a file? How to prevent freehub body damage? I feel like this is poor design. I have two wheelsets that are less than a year old which have damaged freehub bodys due to cogs digging in to the splines on the freehub. It's so bad that I can't remove the cassette spacer without filing down the splines. I had to use  pliers and chain whip to remove the cogs that were embedded in the freehub when trying to remove the cassette.  Most of the damage is on the center of the cassette as the largest 3 cogs are held together by carrier. I'm using a shimano 105 cassette and  Novatec disc hub and Velocity ATB convertible disc hubs (which were somewhat damaged in the same fashion, but not as bad as the novatec).  I'm using this bike for cyclocross, offroad exploring, light singletrack use, Rail trails and some road riding.    How can I prevent this damage to the freehub from happening?