Given the subspace of $L^2$ made by constant functions, characterize its orthogonal complement












1















Let $X={vin L^2([-1,1]):v text{ is constant a.e.}}$ be a subspace of $L^2([-1,1])$, characterize $X^{perp}$.




I don't really know what characterize means, I know that: $X^{perp}={fin L^2([-1,1]):langle f,v rangle=0, forall vin X}$.



$$langle f,v rangle = int^1_{-1}fv dx = vint^1_{-1}f,quad text{since $v$ is constant.}$$



$$vint^1_{-1}f=0quad text{for every $vin X$ iff}quad int^1_{-1}f=0.$$



So $X^{perp}={fin L^2([-1,1]):int^1_{-1}f=0}$.



Is this enough to characterize the set $X^{perp}$ or I need to say something on the orthogonal projections of functions in $L^2([-1,1])$ on $X$ too?






functional-analysis hilbert-spaces







share|cite|improve this question


















  • 1




    Your answer is correct and complete.
    – Kavi Rama Murthy
    Nov 30 at 10:22






  • 1




    complete and correct.
    – DisintegratingByParts
    Nov 30 at 10:38
















1















Let $X={vin L^2([-1,1]):v text{ is constant a.e.}}$ be a subspace of $L^2([-1,1])$, characterize $X^{perp}$.




I don't really know what characterize means, I know that: $X^{perp}={fin L^2([-1,1]):langle f,v rangle=0, forall vin X}$.



$$langle f,v rangle = int^1_{-1}fv dx = vint^1_{-1}f,quad text{since $v$ is constant.}$$



$$vint^1_{-1}f=0quad text{for every $vin X$ iff}quad int^1_{-1}f=0.$$



So $X^{perp}={fin L^2([-1,1]):int^1_{-1}f=0}$.



Is this enough to characterize the set $X^{perp}$ or I need to say something on the orthogonal projections of functions in $L^2([-1,1])$ on $X$ too?






functional-analysis hilbert-spaces







share|cite|improve this question


















  • 1




    Your answer is correct and complete.
    – Kavi Rama Murthy
    Nov 30 at 10:22






  • 1




    complete and correct.
    – DisintegratingByParts
    Nov 30 at 10:38














1












1








1








Let $X={vin L^2([-1,1]):v text{ is constant a.e.}}$ be a subspace of $L^2([-1,1])$, characterize $X^{perp}$.




I don't really know what characterize means, I know that: $X^{perp}={fin L^2([-1,1]):langle f,v rangle=0, forall vin X}$.



$$langle f,v rangle = int^1_{-1}fv dx = vint^1_{-1}f,quad text{since $v$ is constant.}$$



$$vint^1_{-1}f=0quad text{for every $vin X$ iff}quad int^1_{-1}f=0.$$



So $X^{perp}={fin L^2([-1,1]):int^1_{-1}f=0}$.



Is this enough to characterize the set $X^{perp}$ or I need to say something on the orthogonal projections of functions in $L^2([-1,1])$ on $X$ too?






functional-analysis hilbert-spaces







share|cite|improve this question














Let $X={vin L^2([-1,1]):v text{ is constant a.e.}}$ be a subspace of $L^2([-1,1])$, characterize $X^{perp}$.




I don't really know what characterize means, I know that: $X^{perp}={fin L^2([-1,1]):langle f,v rangle=0, forall vin X}$.



$$langle f,v rangle = int^1_{-1}fv dx = vint^1_{-1}f,quad text{since $v$ is constant.}$$



$$vint^1_{-1}f=0quad text{for every $vin X$ iff}quad int^1_{-1}f=0.$$



So $X^{perp}={fin L^2([-1,1]):int^1_{-1}f=0}$.



Is this enough to characterize the set $X^{perp}$ or I need to say something on the orthogonal projections of functions in $L^2([-1,1])$ on $X$ too?






functional-analysis hilbert-spaces




functional-analysis hilbert-spaces






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 30 at 10:15









sound wave

1548




1548








  • 1




    Your answer is correct and complete.
    – Kavi Rama Murthy
    Nov 30 at 10:22






  • 1




    complete and correct.
    – DisintegratingByParts
    Nov 30 at 10:38














  • 1




    Your answer is correct and complete.
    – Kavi Rama Murthy
    Nov 30 at 10:22






  • 1




    complete and correct.
    – DisintegratingByParts
    Nov 30 at 10:38








1




1




Your answer is correct and complete.
– Kavi Rama Murthy
Nov 30 at 10:22




Your answer is correct and complete.
– Kavi Rama Murthy
Nov 30 at 10:22




1




1




complete and correct.
– DisintegratingByParts
Nov 30 at 10:38




complete and correct.
– DisintegratingByParts
Nov 30 at 10:38










1 Answer
1






active

oldest

votes


















1














While the word "characterize" is somewhat ambiguous, since the characterization of a set may be given in many ways (enumeration of elements, "in words", via those satisfying a given proposition, or a union/intersection of some described sets), I am sure whoever has asked this question(person/text) will have expected the answer that you have given i.e. those with integral zero.



Which also means that there is no need to say something on the orthogonal projections of $L^2[-1,1]$ on $X$ (although you may want to think about this : it is not too difficult either).






share|cite|improve this answer





















    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%2f3019926%2fgiven-the-subspace-of-l2-made-by-constant-functions-characterize-its-orthogo%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









    1














    While the word "characterize" is somewhat ambiguous, since the characterization of a set may be given in many ways (enumeration of elements, "in words", via those satisfying a given proposition, or a union/intersection of some described sets), I am sure whoever has asked this question(person/text) will have expected the answer that you have given i.e. those with integral zero.



    Which also means that there is no need to say something on the orthogonal projections of $L^2[-1,1]$ on $X$ (although you may want to think about this : it is not too difficult either).






    share|cite|improve this answer


























      1














      While the word "characterize" is somewhat ambiguous, since the characterization of a set may be given in many ways (enumeration of elements, "in words", via those satisfying a given proposition, or a union/intersection of some described sets), I am sure whoever has asked this question(person/text) will have expected the answer that you have given i.e. those with integral zero.



      Which also means that there is no need to say something on the orthogonal projections of $L^2[-1,1]$ on $X$ (although you may want to think about this : it is not too difficult either).






      share|cite|improve this answer
























        1












        1








        1






        While the word "characterize" is somewhat ambiguous, since the characterization of a set may be given in many ways (enumeration of elements, "in words", via those satisfying a given proposition, or a union/intersection of some described sets), I am sure whoever has asked this question(person/text) will have expected the answer that you have given i.e. those with integral zero.



        Which also means that there is no need to say something on the orthogonal projections of $L^2[-1,1]$ on $X$ (although you may want to think about this : it is not too difficult either).






        share|cite|improve this answer












        While the word "characterize" is somewhat ambiguous, since the characterization of a set may be given in many ways (enumeration of elements, "in words", via those satisfying a given proposition, or a union/intersection of some described sets), I am sure whoever has asked this question(person/text) will have expected the answer that you have given i.e. those with integral zero.



        Which also means that there is no need to say something on the orthogonal projections of $L^2[-1,1]$ on $X$ (although you may want to think about this : it is not too difficult either).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 30 at 11:23









        астон вілла олоф мэллбэрг

        37.2k33376




        37.2k33376






























            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.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • 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.


            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%2f3019926%2fgiven-the-subspace-of-l2-made-by-constant-functions-characterize-its-orthogo%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

            Berounka

            Image
            Berounka.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Caractéristiques Longueur 139  km Bassin 8 861  km 2 Bassin collecteur Elbe Débit moyen ? Cours Source Confluence de la Mže et de la Radbuza · Localisation Pilsen · Altitude 305  m · Coordonnées 49° 45′ 13″ N, 13° 23′ 24″ E Confluence Vltava · Localisation Prague · Altitude 187  m · Coordonnées 49° 59′ 43″ N, 14° 24′ 04″ E Géographie Pays traversés   République tchèque modifier   La Berounka est une rivière de la République tchèque. Elle porte le nom de Mže depuis sa source en Allemagne près de la frontière jusqu'à sa confluence avec la Radbuza à Plzeň. Elle continue ensuite sous le nom de Berounka jusqu'à sa confluence avec la Vltava, près de Prague. La rivière est une destination prisée pour les amateurs de canoë, qui appréc...
            Read more

            Sphinx de Gizeh

            Image
            Pour les articles homonymes, voir Sphinx. Sphinx de Gizeh.mw-parser-output .entete.egypte-antique{background-image:url("//upload.wikimedia.org/wikipedia/commons/c/cc/Picto_infobox_ancient_Egypt.png")} Site d'Égypte antique Le visage du sphinx de Gizeh. Noms en arabe أبو الهول , Abou al-Hôl Localisation Région Basse-Égypte Coordonnées 29° 58′ 31″ nord, 31° 08′ 16″ est Géolocalisation sur la carte : Égypte Sphinx de Gizeh modifier   Le sphinx de Gizeh est la statue thérianthrope qui se dresse devant les grandes pyramides du plateau de Gizeh, en Basse-Égypte. Sculpture monumentale monolithique la plus grande du monde [ 1 ] avec 73,5 mètres de longueur, 14 mètres de largeur et 20,22 mètres de hauteur, elle représente un sphinx couchant. Réalisée vers -2500, elle serait attribuée à Djédefrê, l'un des pharaons de la IV e  dynastie, représentant son père le pharaon Khéops. Sommaire 1 Étymologie 2 Desc...
            Read more

            Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

            Image
            3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a...
            Read more