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

666216




666216












  • $begingroup$
    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.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    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

        59k42056




        59k42056






























            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

            Bressuire

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more

            Cabo Verde

            Image
              Nota: Cabo-verdiano redireciona para este artigo. Para a língua falada em Cabo Verde, veja crioulo cabo-verdiano. Para outros significados, veja Cabo Verde (desambiguação). Coordenadas: 14° 55' N 23° 31' O Cabo Verde República de Cabo Verde Bandeira Brasão de Armas Lema: Unidade, Trabalho, Progresso Hino nacional: Cântico da Liberdade Gentílico: cabo-verdiano [ 1 ] Capital Praia 14° 55' N 23° 31' O Cidade mais populosa Praia Língua oficial Português Governo República unitária semipresidencialista [ 2 ]  - Presidente Jorge Carlos Fonseca  - Primeiro-ministro Ulisses Correia e Silva História    - Descoberta 1460   - Independência 00 • de Portugal 5 de Julho de 1975   - Multipartidarismo 13 de Janeiro de 1990  Área    - Total 4 033 km² (146.º)  - Água (%) 0,08 População    - Estimativa para 2017 560 899 [ 3 ] hab. (173.º) ...
            Read more

            Gyllenstierna

            Image
            .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid ...
            Read more