The definition of a Hilbert space is a complete inner product space. I am struggling to imagine what an incomplete inner product space would look like, especially since inner products by definition must be maps on vector spaces whose scalar fields are either $\mathbb R$ or $\mathbb C$. Since $\mathbb R$ and $\mathbb C$ are complete and inner products are (sesqui)linear, it seems that any inner product must be complete. Is the definition of Hilbert space redundant?  In short, what's an example of an inner product space that is not complete?  It seems one of the following must be the case:   All inner product spaces are complete, and the definition of Hilbert space is redundant. Inner products can be defined on more general fields than $\mathbb R$ and $\mathbb C$, for example $\mathbb Q$ and other subfields of $\mathbb R$. The definition of Hilbert space simply rules out these incomplete subfields from consideration. There exist genuine inner product spaces over $\mathbb R$ and/or $\mathbb C$ that are not complete, which would be the most interesting and unexpected situation.
That is, what are some good examples of vector spaces which are inner product spaces but in which not every Cauchy sequence converges?
problems using apa6e with biblatex-apa As in  I'm trying to use apa6e and biblatex-apa together and having problems.  Before I arrived at these problems, over the course of many hours of much reading, I first managed to get BibTeX working as expected in a simple input file (by modifying sample2e.tex). Then I managed to get apa6e working as expected based on the author's example in apa6e.pdf (which uses apacite). But when I try to get apa6e working with biblatex-apa, I end up with problems.  Here's my input file:  \documentclass{apa6e}  \usepackage[american]{babel} \usepackage{csquotes}  % biblatex \usepackage[style=apa]{biblatex} \bibliography{biblatex-apa-test-references}  \title{An Example Document} \shorttitle{Psychology: the end} \author{Silly Name}  \authornote{\dots} \abstract{\dots}  \begin{document}  \maketitle  This is an example input\cite{7.01:2} file.  Comparing it with the output it generates can show you\cite{7.01:3a} how to produce a simple document of your own.  \printbibliography  \end{document}   Attempting to typeset this in TeXShop by running LaTeX, then BiBTeX (which based on console messages, does seem to correctly call biblatex rather than BiBTeX), then LaTeX again, I get the following error message which I am having great difficulty (no doubt due to my lack of experience with TeX and co.) decoding.  (./test6.aux) (/usr/local/texlive/2011/texmf-dist/tex/latex/psnfss/ot1ptm.fd) *geometry* driver: auto-detecting *geometry* detected driver: pdftex (./test6.bbl) Underfull \hbox (badness 10000) in paragraph at lines 18--18  []\OT1/ptm/m/n/12 An Ex-am-ple Doc-u-ment   Underfull \hbox (badness 10000) in paragraph at lines 18--18  []\OT1/ptm/m/n/12 Silly Name   Underfull \hbox (badness 10000) in paragraph at lines 18--18  []\OT1/ptm/m/n/12 Author Note  [1{/usr/local/texlive/2011/texmf-var/fonts/map/pdftex/updmap/pdftex.map}] Underfull \hbox (badness 10000) in paragraph at lines 18--18  []\OT1/ptm/m/n/12 Abstract  [2] Underfull \hbox (badness 10000) in paragraph at lines 18--18  []\OT1/ptm/m/n/12 An Ex-am-ple Doc-u-ment   Underfull \hbox (badness 10000) in paragraph at lines 25--25  []\OT1/ptm/b/n/12 References  ./test6.tex:25: Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra  (/usr/local/texlive/2011/texmf-dist/tex/latex/psnfss/ot1pcr.fd) (/usr/local/texlive/2011/texmf-dist/tex/latex/psnfss/ot1ztmcm.fd) (/usr/local/texlive/2011/texmf-dist/tex/latex/psnfss/omlztmcm.fd) (/usr/local/texlive/2011/texmf-dist/tex/latex/psnfss/omsztmcm.fd) (/usr/local/texlive/2011/texmf-dist/tex/latex/psnfss/omxztmcm.fd) Overfull \hbox (8.03372pt too wide) in paragraph at lines 25--25 [][]  \OT1/ptm/m/n/12 Gilbert, D. G., Mc-Cler-non, J. F., Ra-bi-novich, N. E.,  Sugai, C., Plath, L. C., As-gaard, G., ... Botros,  ./test6.tex:25: Undefined control sequence. &lt;argument&gt; \mkbibdateapalongextra                                    {year}{month}{day}\iffieldundef {endyear}{... l.25   I tried many different sources in several different *.bib files before eventually trying the .bib database shown above and included with biblatex-apa. All trials gave a similar error.  Can someone help? I'm impressed that the poster above was able to do anything with apa6e and biblatex-apa, and if I only had her problem of having the reference page title show up in bold, I'd be really happy.  If you need to know, my tools are from a fresh installation of MacTeX that I immediately updated using the TeX Live Utility. I've checked my versions for apa6e (v0.3) and biblatex-apa (v4.2), and I think these are current. While reading a question here whose answer was Stephen Baxter's &quot;Light of Other Days&quot; book, it reminded me of a short story I read with a similar idea. Probably read at least 30-40 years ago, in English. The fact that the Government has invented a device that can view the past is made public (by a whistleblower, I believe). Historians go nuts besieging the government to look at their favorite time periods to reveal the truth about various historical &quot;facts&quot; and or &quot;mysteries&quot; (Who shot JFK? Who was Jack the Ripper? Was Helen of Troy really that beautiful? Did Jesus exist? etc, etc.) And the government is stonewalls most of them and claims it's expensive to operate, hard to control to get exact time/location, etc. So someone (maybe the whistleblower?), knowing it's possible, does math/construction and builds his own version, and realizes it's easy to make, cheap to operate, and easy to use. So he makes plans to stick it to the government by releasing the info. Maybe he tries to force the government to release the info, or they realize he has made it, so they grab him and try to convince him to stop. But he is not convinced until they tell him, it can't see back very far (maybe just a few days/weeks, because it needs exponentially more power the further you look back.) Finally the agents tell him: &quot;Don't you see, you can use it to see what's happening just a few minutes ago, or even now. Your company can't make widgets? Zoom in on your competitors and see how they make them. Think your wife is fooling around on you? Zoom in on her and track her to see if she is. Your pretty neighbor is taking a bath now? Zoom in and watch her... These devices mean no one will have any privacy now.&quot; At which point he tells them it's already too late. They kept him there so long that the letters he mailed will already be delivered to various newspapers, scientists, historians, etc. How to show distance between boundary of an open set and a compact subset is positive? Let $n \in \mathbb N$ and $U\subset \mathbb R^n$ be an open set.Let $K\subset U$ be a compact subset of $U$. Then How can we show that distance between $K$ and $\partial U$ is positive?   Please, someone, give some hints. to solve this.  Thank you. Can we clarify to the OP that their question is not yet closed and the duplicates are only suggestions? For every set $A$, the empty set is a subset of $A$. The empty set is a set. Therefore, the empty set has a cardinality $\geq 1\ldots$ Most of what I have read about retry operations is that they are vaguely complicated and indepth. I have written the code below which seems to accomplish this without too much confusion - I'm wondering if there are any pitfalls I am not seeing here.   The class uses a top-level schedulable class, which calls a queueable class to make the callout, and if the callout is unsuccessful, it just schedules the top level class again five minutes later. After testing it appears this works indefinitely.   A simple string token, assigned at the first invocation, is also passed between the classes, and I have tested and found that the token does persist, even when runs are scheduled for a later time.   Thanks for helpful thoughts!  public class SelfServe implements Schedulable {      public String Token = null;      public void execute(SchedulableContext ctx) {                  RetrieveMachineDetails Job = new RetrieveMachineDetails();             Job.Token = Token;             System.enqueueJob(Job);      }           public class RetrieveMachineDetails implements Queueable, Database.AllowsCallouts {          public String Token = null;          public void execute(QueueableContext context){                                            Http http = new Http();             HttpRequest request = new HttpRequest();                             request.setEndpoint('https://app.endpoint.com/api');                     request.setMethod('POST');                     request.setHeader('Content-Type', 'application/json;charset=UTF-8');                                      request.setBody('{"query":  "$message_=\'METRIC\' &amp;&amp; $tier=\'customer\' "}');                                                                                                                                                                                    HttpResponse response = http.send(request);                            if (response.getStatusCode() == 200) {                 //do stuff                  }              else { //retry                  DateTime Dt = system.now().addMinutes(5);                                String day = string.valueOf(Dt.day());                                        String month = string.valueOf(Dt.month());                                          String hour = string.valueOf(Dt.hour());                                       String minute = string.valueOf(Dt.minute());                                                       String year = string.valueOf(Dt.year());                                   String ScheduledTime = '0 ' + minute + ' ' + hour + ' ' + day + ' ' + month + ' ?' + ' ' + year;                  SelfServe Job = new SelfServe();                     Job.Token = Token;                 System.schedule('SelfServe Job'+Dt, ScheduledTime, Job);              }         }        }  } Prime factorization knowing n and Euler's function My task is to factorize $n$ knowing it has two factors and $\varphi(n)$. How can I do it?  Thanks for any advice. I failed to open a lock. Now what? The Unix environment has some excellent tools for parsing text in various forms. However, of late, the data is not in the traditional (historical) formats (CSV, TSV, record-based or some other delimiter-based) it used to be before. Data these days is exchanged in structured formats like XML/JSON.  I know there are some good tools like sed, awk and Perl which can chew down nearly any form of data out there. However, to work with this sort of structured data, often one has to write a complete program, and, given the little time available to extract information, one has to sit down and figure out the whole logic of what one wants to query and put it down programmatically. Sometimes this is not OK – basically because the information extracted from those files acts as inputs for further work; also because of the time it takes to search for the appropriate solution and code it up. A command line tool is needed with sufficient switches to find, query and dump data.  I'm looking for tools that take a XML/JSON or other forms of structured data and dump it into other formats like csv, etc., so that from there one could use other commands to get any information out of it.  Are there any command line utilities you know of which do this kind of a job? Are there already awk/Perl scripts available to this? I bet the automatic response to my question would be "Bell's theorem" and of course I am not disputing Bell's proof. I am however uncertain of one of his assumptions.  The so called "no conspiracy" assumption states that we somehow posses this magical thing called free will which makes us able to break free from the causal chain of events that make us measure the $y$-axis instead of the $x$-axis and so on in experiments. Obviously this makes little sense once reviewed under the light of logic. What this naturally implies is that, if a theory is fully deterministic, it is instantly super-deterministic.  Simon Kochen and John Conway published a theorem in 2009, , which simply-put states: either everything is deterministic (super-deterministic) or every particle has free will.  Among the few who accept this to the fullest extent is Gerard 't Hooft. He has proposed that what we call sub-atomic particles are really just templates and that there is a realm beneath the quantum where the true fabric of reality is "hidden". By accepting determinism he escapes Bell's theorem (due to not having to accept the "no conspiracy" assumption) and can construct a local, deterministic and realist hypothesis of reality. In Gerard 't Hooft's model there is cellular automata somewhere near the Planck scale which gives rise to the "emergent quantum mechanics".  This means that there exist no superpositions in objective reality, the cat is always either dead or alive, no collapse of wave functions, no branching universes, non-locality or retro-causality.  The big question for me is basically: Why do people have such a hard time accepting determinism? Some will object and say "oh but it's SUPER-determinism", but that makes no sense whatsoever, either everything is determined (determinism) or only somethings are (quasi-determinism). By accepting it we escape Bell's theorem. Bell himself was well aware of this and mentioned it a few times in interviews in the 80's.  I know some are concerned that if we accept complete determinism we can no longer do science because it's all a HUGE CONSPIRACY. Gerard 't Hooft has answered critics who bring up this here: , section 6.  Additionally, another author who has taken a deeper look into the worries of "conspiracies" in QM interpretations is Peter J. Lewis in his paper  (also ).  So to all of you who dismiss local hidden variables I ask: Why? "Geometric" or "Geometrical"? I have read the excellent answers to  - my question is specifically about usage.   Is there a best practice for deciding between the variants geometric and geometrical in writing? A  shows that geometrical was more popular until about 1965, when they switched. What grammatical rules, if any, would be relevant here? What does the fourth number in the release version mean? I'm working on a new edition of a book which uses a very unusual style of bibliography: Entries are sorted in author-year fashion, but then numbered per author or group of authors, e.g.:  Milnor, J.:    1. Construction of universal bundles: I, Ann. Math., (2)63:272-284 (1956).    2. Construction of universal bundles: II, Ann. Math., (2)63: 430-436 (1956).    3. On manifolds homeomorphic to the 7-sphere, Ann. Math., (2)64: 399-405       (1956).    4. The theory of characteristic classes, mimeographed, Princeton University,       Princeton, N.J., 1957. Milnor, J., and M. A. Kervaire:    1. Bernoulli numbers, homotopy groups, and a theorem of Rohlin, Proc. Intern.       Congr. Math., 1958. Milnor, J., and E. Spanier:    1. Two remarks on fibre homotopy type, Pacific J. Math., 10: 585-590 (1960).   Citations are formatted as Milnor [3], and so on.  Is there any bibtex style for something strange as this? Or can this type of bibliography be created with biblatex without too much coding? Cant find the solution when sorting an array in JavaScript How to test with Math.random in JavaScript? Sum two dates in Java How do I make Endermen spawn more frequently? I'm playing vanilla Minecraft, and am getting ready to head to the end.  I need Eyes of Ender (and hence Ender Pearls), and in my main hunting grounds (a large desert area) as little as one Enderman may spawn per night.  Is there any way to "promote" Enderman spawning so I can get pearls more quickly? Proving that $a \mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb Z$ Given $m \in \mathbb{Z}$, let $m\mathbb{Z}$ denote the set of integer multiples of $m$, i.e. $m\mathbb{Z} := \{mk\mid k \in \mathbb{Z}\}$.   Now let $a,b \in \mathbb{Z}$ with $a,b$ not both $0$. Prove that $a\mathbb{Z} \cap b\mathbb{Z} = \operatorname{lcm}(a,b)\mathbb{Z}$.   I am trying to write a proof for this, but I am unsure of what method to use. Also I am confused by $mk\mid k$, because wouldn't $m=1$ for this to be true. reason for exec in wrapper scripts Count occurrences of each element in a List[List[T]] in Scala