What does this notation means? (proof of Picard–Lindelöf theorem in wikipedia)












0












$begingroup$


I was read a detailed proof of Picard–Lindelöf theorem. And, in this article of Wikipedia I found a notation some weird.
I saw that:
$$overline {I_{a}(t_{0})} = [t_0-a,t_0+a]$$
$$overline {B_{b}(y_{0})} = [y_0-b,y_0+b]$$
This is al normal, it is a definition.
But after, I saw:



$$displaystyle Gamma : {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )longrightarrow {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )$$



So, What does this line over the letter mean? It is an error? Because I had the idea that it is:



$$displaystyle Gamma : {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)longrightarrow {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)$$



The other thing is, according to me, the theorem should have a proof with $overline {B_{b}(y_{0})} subseteq mathbb{R}^n$, but $overline {B_{b}(y_{0})}$ is a interval in this proof.



Sorry for my english.






general-topology ordinary-differential-equations notation







share|cite|improve this question











$endgroup$












  • $begingroup$
    I don't see any 'hat' anywhere.
    $endgroup$
    – Kavi Rama Murthy
    Dec 29 '18 at 23:56










  • $begingroup$
    I wanted to say "line over the letter", thank you
    $endgroup$
    – El borito
    Dec 30 '18 at 7:36










  • $begingroup$
    An overline is usually to denote closure. For example if $I = (a, b)$ is an open interval, then it's closure is $overline{I} = [a, b]$.
    $endgroup$
    – Pratyush Sarkar
    Dec 30 '18 at 7:51












  • $begingroup$
    So, there is a possibility that it is an error?
    $endgroup$
    – El borito
    Dec 30 '18 at 8:23










  • $begingroup$
    The proof is for $n=1$, so a closed ball reduces to a compact interval.
    $endgroup$
    – user539887
    Dec 30 '18 at 9:04


















0












$begingroup$


I was read a detailed proof of Picard–Lindelöf theorem. And, in this article of Wikipedia I found a notation some weird.
I saw that:
$$overline {I_{a}(t_{0})} = [t_0-a,t_0+a]$$
$$overline {B_{b}(y_{0})} = [y_0-b,y_0+b]$$
This is al normal, it is a definition.
But after, I saw:



$$displaystyle Gamma : {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )longrightarrow {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )$$



So, What does this line over the letter mean? It is an error? Because I had the idea that it is:



$$displaystyle Gamma : {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)longrightarrow {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)$$



The other thing is, according to me, the theorem should have a proof with $overline {B_{b}(y_{0})} subseteq mathbb{R}^n$, but $overline {B_{b}(y_{0})}$ is a interval in this proof.



Sorry for my english.






general-topology ordinary-differential-equations notation







share|cite|improve this question











$endgroup$












  • $begingroup$
    I don't see any 'hat' anywhere.
    $endgroup$
    – Kavi Rama Murthy
    Dec 29 '18 at 23:56










  • $begingroup$
    I wanted to say "line over the letter", thank you
    $endgroup$
    – El borito
    Dec 30 '18 at 7:36










  • $begingroup$
    An overline is usually to denote closure. For example if $I = (a, b)$ is an open interval, then it's closure is $overline{I} = [a, b]$.
    $endgroup$
    – Pratyush Sarkar
    Dec 30 '18 at 7:51












  • $begingroup$
    So, there is a possibility that it is an error?
    $endgroup$
    – El borito
    Dec 30 '18 at 8:23










  • $begingroup$
    The proof is for $n=1$, so a closed ball reduces to a compact interval.
    $endgroup$
    – user539887
    Dec 30 '18 at 9:04
















0












0








0





$begingroup$


I was read a detailed proof of Picard–Lindelöf theorem. And, in this article of Wikipedia I found a notation some weird.
I saw that:
$$overline {I_{a}(t_{0})} = [t_0-a,t_0+a]$$
$$overline {B_{b}(y_{0})} = [y_0-b,y_0+b]$$
This is al normal, it is a definition.
But after, I saw:



$$displaystyle Gamma : {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )longrightarrow {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )$$



So, What does this line over the letter mean? It is an error? Because I had the idea that it is:



$$displaystyle Gamma : {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)longrightarrow {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)$$



The other thing is, according to me, the theorem should have a proof with $overline {B_{b}(y_{0})} subseteq mathbb{R}^n$, but $overline {B_{b}(y_{0})}$ is a interval in this proof.



Sorry for my english.






general-topology ordinary-differential-equations notation







share|cite|improve this question











$endgroup$




I was read a detailed proof of Picard–Lindelöf theorem. And, in this article of Wikipedia I found a notation some weird.
I saw that:
$$overline {I_{a}(t_{0})} = [t_0-a,t_0+a]$$
$$overline {B_{b}(y_{0})} = [y_0-b,y_0+b]$$
This is al normal, it is a definition.
But after, I saw:



$$displaystyle Gamma : {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )longrightarrow {mathcal {C}} ( I_{a}(t_{0}) , B_{b}(y_{0}) )$$



So, What does this line over the letter mean? It is an error? Because I had the idea that it is:



$$displaystyle Gamma : {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)longrightarrow {mathcal {C}} left( overline {I_{a}(t_{0})} , overline {B_{b}(y_{0})} right)$$



The other thing is, according to me, the theorem should have a proof with $overline {B_{b}(y_{0})} subseteq mathbb{R}^n$, but $overline {B_{b}(y_{0})}$ is a interval in this proof.



Sorry for my english.






general-topology ordinary-differential-equations notation




general-topology ordinary-differential-equations notation






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 30 '18 at 7:34







El borito

















asked Dec 29 '18 at 23:09









El boritoEl borito

664216




664216












  • $begingroup$
    I don't see any 'hat' anywhere.
    $endgroup$
    – Kavi Rama Murthy
    Dec 29 '18 at 23:56










  • $begingroup$
    I wanted to say "line over the letter", thank you
    $endgroup$
    – El borito
    Dec 30 '18 at 7:36










  • $begingroup$
    An overline is usually to denote closure. For example if $I = (a, b)$ is an open interval, then it's closure is $overline{I} = [a, b]$.
    $endgroup$
    – Pratyush Sarkar
    Dec 30 '18 at 7:51












  • $begingroup$
    So, there is a possibility that it is an error?
    $endgroup$
    – El borito
    Dec 30 '18 at 8:23










  • $begingroup$
    The proof is for $n=1$, so a closed ball reduces to a compact interval.
    $endgroup$
    – user539887
    Dec 30 '18 at 9:04




















  • $begingroup$
    I don't see any 'hat' anywhere.
    $endgroup$
    – Kavi Rama Murthy
    Dec 29 '18 at 23:56










  • $begingroup$
    I wanted to say "line over the letter", thank you
    $endgroup$
    – El borito
    Dec 30 '18 at 7:36










  • $begingroup$
    An overline is usually to denote closure. For example if $I = (a, b)$ is an open interval, then it's closure is $overline{I} = [a, b]$.
    $endgroup$
    – Pratyush Sarkar
    Dec 30 '18 at 7:51












  • $begingroup$
    So, there is a possibility that it is an error?
    $endgroup$
    – El borito
    Dec 30 '18 at 8:23










  • $begingroup$
    The proof is for $n=1$, so a closed ball reduces to a compact interval.
    $endgroup$
    – user539887
    Dec 30 '18 at 9:04


















$begingroup$
I don't see any 'hat' anywhere.
$endgroup$
– Kavi Rama Murthy
Dec 29 '18 at 23:56




$begingroup$
I don't see any 'hat' anywhere.
$endgroup$
– Kavi Rama Murthy
Dec 29 '18 at 23:56












$begingroup$
I wanted to say "line over the letter", thank you
$endgroup$
– El borito
Dec 30 '18 at 7:36




$begingroup$
I wanted to say "line over the letter", thank you
$endgroup$
– El borito
Dec 30 '18 at 7:36












$begingroup$
An overline is usually to denote closure. For example if $I = (a, b)$ is an open interval, then it's closure is $overline{I} = [a, b]$.
$endgroup$
– Pratyush Sarkar
Dec 30 '18 at 7:51






$begingroup$
An overline is usually to denote closure. For example if $I = (a, b)$ is an open interval, then it's closure is $overline{I} = [a, b]$.
$endgroup$
– Pratyush Sarkar
Dec 30 '18 at 7:51














$begingroup$
So, there is a possibility that it is an error?
$endgroup$
– El borito
Dec 30 '18 at 8:23




$begingroup$
So, there is a possibility that it is an error?
$endgroup$
– El borito
Dec 30 '18 at 8:23












$begingroup$
The proof is for $n=1$, so a closed ball reduces to a compact interval.
$endgroup$
– user539887
Dec 30 '18 at 9:04






$begingroup$
The proof is for $n=1$, so a closed ball reduces to a compact interval.
$endgroup$
– user539887
Dec 30 '18 at 9:04












1 Answer
1






active

oldest

votes


















0












$begingroup$

Yes, the wikipedia entry is sub-optimal. There should be mention of a domain $cal D$ of $f$ in the first statement in the intro, later that the cylinder $bar I_atimes bar B_b$ has to be a sub-set of $cal D$.



And you need the closed sets, first for compactness to have a maximum $M$ of $f$ over the cylinder and later for completeness of the function space as you apply the Banach fixed-point theorem.



(Btw., there is no need to invoke Grönwall, as the BFS already supplies the uniqueness of the fixed point.)




In short, as always with the things written in wikipedia, look for other sources (more than one). For this topic, preferably find text books, as the proofs there are usually more carefully composed.






share|cite|improve this answer









$endgroup$














    Your Answer








    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056352%2fwhat-does-this-notation-means-proof-of-picard-lindel%25c3%25b6f-theorem-in-wikipedia%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    0












    $begingroup$

    Yes, the wikipedia entry is sub-optimal. There should be mention of a domain $cal D$ of $f$ in the first statement in the intro, later that the cylinder $bar I_atimes bar B_b$ has to be a sub-set of $cal D$.



    And you need the closed sets, first for compactness to have a maximum $M$ of $f$ over the cylinder and later for completeness of the function space as you apply the Banach fixed-point theorem.



    (Btw., there is no need to invoke Grönwall, as the BFS already supplies the uniqueness of the fixed point.)




    In short, as always with the things written in wikipedia, look for other sources (more than one). For this topic, preferably find text books, as the proofs there are usually more carefully composed.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Yes, the wikipedia entry is sub-optimal. There should be mention of a domain $cal D$ of $f$ in the first statement in the intro, later that the cylinder $bar I_atimes bar B_b$ has to be a sub-set of $cal D$.



      And you need the closed sets, first for compactness to have a maximum $M$ of $f$ over the cylinder and later for completeness of the function space as you apply the Banach fixed-point theorem.



      (Btw., there is no need to invoke Grönwall, as the BFS already supplies the uniqueness of the fixed point.)




      In short, as always with the things written in wikipedia, look for other sources (more than one). For this topic, preferably find text books, as the proofs there are usually more carefully composed.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Yes, the wikipedia entry is sub-optimal. There should be mention of a domain $cal D$ of $f$ in the first statement in the intro, later that the cylinder $bar I_atimes bar B_b$ has to be a sub-set of $cal D$.



        And you need the closed sets, first for compactness to have a maximum $M$ of $f$ over the cylinder and later for completeness of the function space as you apply the Banach fixed-point theorem.



        (Btw., there is no need to invoke Grönwall, as the BFS already supplies the uniqueness of the fixed point.)




        In short, as always with the things written in wikipedia, look for other sources (more than one). For this topic, preferably find text books, as the proofs there are usually more carefully composed.






        share|cite|improve this answer









        $endgroup$



        Yes, the wikipedia entry is sub-optimal. There should be mention of a domain $cal D$ of $f$ in the first statement in the intro, later that the cylinder $bar I_atimes bar B_b$ has to be a sub-set of $cal D$.



        And you need the closed sets, first for compactness to have a maximum $M$ of $f$ over the cylinder and later for completeness of the function space as you apply the Banach fixed-point theorem.



        (Btw., there is no need to invoke Grönwall, as the BFS already supplies the uniqueness of the fixed point.)




        In short, as always with the things written in wikipedia, look for other sources (more than one). For this topic, preferably find text books, as the proofs there are usually more carefully composed.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 30 '18 at 10:31









        LutzLLutzL

        60.9k42157




        60.9k42157






























            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3056352%2fwhat-does-this-notation-means-proof-of-picard-lindel%25c3%25b6f-theorem-in-wikipedia%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Bundesstraße 106

            Image
            Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück
            Read more

            Verónica Boquete

            Image
            Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl
            Read more

            Ida-Boy-Ed-Garten

            Image
            Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich
            Read more