Uniform Boundedness principle for bounded linear maps from Frechet Space into a Banach Space












2












$begingroup$


I am looking for a proof of the uniform boundedness principle where the domain is a Frechet space, instead of the usual setting of a Banach Space.



This is used in proving the space of tempered distributions is complete but I can't find a proof of it anywhere.



When I try to prove it myself I get stuck on the final part(which uses the scaling property of linear maps).



Does anyone have a proof that they could share?






real-analysis functional-analysis distribution-theory







share|cite|improve this question











$endgroup$












  • $begingroup$
    Doesn't the standard proof via the Baire category theorem just work?
    $endgroup$
    – tomasz
    Jan 17 '17 at 0:46










  • $begingroup$
    The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 0:49












  • $begingroup$
    HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 1:05
















2












$begingroup$


I am looking for a proof of the uniform boundedness principle where the domain is a Frechet space, instead of the usual setting of a Banach Space.



This is used in proving the space of tempered distributions is complete but I can't find a proof of it anywhere.



When I try to prove it myself I get stuck on the final part(which uses the scaling property of linear maps).



Does anyone have a proof that they could share?






real-analysis functional-analysis distribution-theory







share|cite|improve this question











$endgroup$












  • $begingroup$
    Doesn't the standard proof via the Baire category theorem just work?
    $endgroup$
    – tomasz
    Jan 17 '17 at 0:46










  • $begingroup$
    The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 0:49












  • $begingroup$
    HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 1:05














2












2








2


2



$begingroup$


I am looking for a proof of the uniform boundedness principle where the domain is a Frechet space, instead of the usual setting of a Banach Space.



This is used in proving the space of tempered distributions is complete but I can't find a proof of it anywhere.



When I try to prove it myself I get stuck on the final part(which uses the scaling property of linear maps).



Does anyone have a proof that they could share?






real-analysis functional-analysis distribution-theory







share|cite|improve this question











$endgroup$




I am looking for a proof of the uniform boundedness principle where the domain is a Frechet space, instead of the usual setting of a Banach Space.



This is used in proving the space of tempered distributions is complete but I can't find a proof of it anywhere.



When I try to prove it myself I get stuck on the final part(which uses the scaling property of linear maps).



Does anyone have a proof that they could share?






real-analysis functional-analysis distribution-theory




real-analysis functional-analysis distribution-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 17 '17 at 0:17









spaceisdarkgreen

34k21754




34k21754










asked Jan 16 '17 at 23:44









FourierFluxFourierFlux

987




987












  • $begingroup$
    Doesn't the standard proof via the Baire category theorem just work?
    $endgroup$
    – tomasz
    Jan 17 '17 at 0:46










  • $begingroup$
    The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 0:49












  • $begingroup$
    HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 1:05


















  • $begingroup$
    Doesn't the standard proof via the Baire category theorem just work?
    $endgroup$
    – tomasz
    Jan 17 '17 at 0:46










  • $begingroup$
    The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 0:49












  • $begingroup$
    HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range.
    $endgroup$
    – FourierFlux
    Jan 17 '17 at 1:05
















$begingroup$
Doesn't the standard proof via the Baire category theorem just work?
$endgroup$
– tomasz
Jan 17 '17 at 0:46




$begingroup$
Doesn't the standard proof via the Baire category theorem just work?
$endgroup$
– tomasz
Jan 17 '17 at 0:46












$begingroup$
The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me.
$endgroup$
– FourierFlux
Jan 17 '17 at 0:49






$begingroup$
The problem the last part relies upon the scaling property of a linear map(pulling out a scaler). I admit I'm not well versed in Frechet spaces though - if I'm wrong please correct me.
$endgroup$
– FourierFlux
Jan 17 '17 at 0:49














$begingroup$
HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range.
$endgroup$
– FourierFlux
Jan 17 '17 at 1:05




$begingroup$
HAHAHA, Ok I see. Being stupid I was thinking the norm was of the domain and not the range.
$endgroup$
– FourierFlux
Jan 17 '17 at 1:05










1 Answer
1






active

oldest

votes


















1












$begingroup$

Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X rightarrow Y$ be continuous linear and $q: Y rightarrow [0,infty)$ be a continuous semi-norm. If $sup {q(T_a x):ain A} <infty$ (pointwise bounded), then $sup {q(T_a x):ain A}$ is a continuous semi-norm (uniformly bounded/equicontinuous).



To prove this, let
$$E_n = {xin X: q(T_a x) le n, forall a}$$
By pointwise boundedness, we see that $E_n nearrow X$. Since $E_n$ are closed and $X$ is a Baire space, we see that there exists $E_n$ with an interior point $x$. Since $X$ is Frechet, its topology is generated by countable semi-norms $p_n$ and thus there exists $N,r$ such that
$$
x + bigcap_{k=1}^N {p_k < r} subseteq E_n
$$
Hence,
$$
bigcap_{k=1}^N {p_k < 2r} subseteq E_n -E_n subseteq E_{2n}
$$
Let
$$
p(x) = frac{1}{r} sum_{k=1}^N p_k (x)
$$
Hence,
$$
0in {p < 1} subseteq {sup {q(T_a x):ain A} <3n}
$$
Since $p$ is continuous, we see that $0$ is an interior point of ${sup {q(T_a x):ain A} <3n} $ and thus $sup {q(T_a x):ain A}$ is a continuous semi-norm.






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%2f2100943%2funiform-boundedness-principle-for-bounded-linear-maps-from-frechet-space-into-a%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












    $begingroup$

    Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X rightarrow Y$ be continuous linear and $q: Y rightarrow [0,infty)$ be a continuous semi-norm. If $sup {q(T_a x):ain A} <infty$ (pointwise bounded), then $sup {q(T_a x):ain A}$ is a continuous semi-norm (uniformly bounded/equicontinuous).



    To prove this, let
    $$E_n = {xin X: q(T_a x) le n, forall a}$$
    By pointwise boundedness, we see that $E_n nearrow X$. Since $E_n$ are closed and $X$ is a Baire space, we see that there exists $E_n$ with an interior point $x$. Since $X$ is Frechet, its topology is generated by countable semi-norms $p_n$ and thus there exists $N,r$ such that
    $$
    x + bigcap_{k=1}^N {p_k < r} subseteq E_n
    $$
    Hence,
    $$
    bigcap_{k=1}^N {p_k < 2r} subseteq E_n -E_n subseteq E_{2n}
    $$
    Let
    $$
    p(x) = frac{1}{r} sum_{k=1}^N p_k (x)
    $$
    Hence,
    $$
    0in {p < 1} subseteq {sup {q(T_a x):ain A} <3n}
    $$
    Since $p$ is continuous, we see that $0$ is an interior point of ${sup {q(T_a x):ain A} <3n} $ and thus $sup {q(T_a x):ain A}$ is a continuous semi-norm.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X rightarrow Y$ be continuous linear and $q: Y rightarrow [0,infty)$ be a continuous semi-norm. If $sup {q(T_a x):ain A} <infty$ (pointwise bounded), then $sup {q(T_a x):ain A}$ is a continuous semi-norm (uniformly bounded/equicontinuous).



      To prove this, let
      $$E_n = {xin X: q(T_a x) le n, forall a}$$
      By pointwise boundedness, we see that $E_n nearrow X$. Since $E_n$ are closed and $X$ is a Baire space, we see that there exists $E_n$ with an interior point $x$. Since $X$ is Frechet, its topology is generated by countable semi-norms $p_n$ and thus there exists $N,r$ such that
      $$
      x + bigcap_{k=1}^N {p_k < r} subseteq E_n
      $$
      Hence,
      $$
      bigcap_{k=1}^N {p_k < 2r} subseteq E_n -E_n subseteq E_{2n}
      $$
      Let
      $$
      p(x) = frac{1}{r} sum_{k=1}^N p_k (x)
      $$
      Hence,
      $$
      0in {p < 1} subseteq {sup {q(T_a x):ain A} <3n}
      $$
      Since $p$ is continuous, we see that $0$ is an interior point of ${sup {q(T_a x):ain A} <3n} $ and thus $sup {q(T_a x):ain A}$ is a continuous semi-norm.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X rightarrow Y$ be continuous linear and $q: Y rightarrow [0,infty)$ be a continuous semi-norm. If $sup {q(T_a x):ain A} <infty$ (pointwise bounded), then $sup {q(T_a x):ain A}$ is a continuous semi-norm (uniformly bounded/equicontinuous).



        To prove this, let
        $$E_n = {xin X: q(T_a x) le n, forall a}$$
        By pointwise boundedness, we see that $E_n nearrow X$. Since $E_n$ are closed and $X$ is a Baire space, we see that there exists $E_n$ with an interior point $x$. Since $X$ is Frechet, its topology is generated by countable semi-norms $p_n$ and thus there exists $N,r$ such that
        $$
        x + bigcap_{k=1}^N {p_k < r} subseteq E_n
        $$
        Hence,
        $$
        bigcap_{k=1}^N {p_k < 2r} subseteq E_n -E_n subseteq E_{2n}
        $$
        Let
        $$
        p(x) = frac{1}{r} sum_{k=1}^N p_k (x)
        $$
        Hence,
        $$
        0in {p < 1} subseteq {sup {q(T_a x):ain A} <3n}
        $$
        Since $p$ is continuous, we see that $0$ is an interior point of ${sup {q(T_a x):ain A} <3n} $ and thus $sup {q(T_a x):ain A}$ is a continuous semi-norm.






        share|cite|improve this answer











        $endgroup$



        Let $X$ be Frechet and $Y$ be locally convex. Let $T_a: X rightarrow Y$ be continuous linear and $q: Y rightarrow [0,infty)$ be a continuous semi-norm. If $sup {q(T_a x):ain A} <infty$ (pointwise bounded), then $sup {q(T_a x):ain A}$ is a continuous semi-norm (uniformly bounded/equicontinuous).



        To prove this, let
        $$E_n = {xin X: q(T_a x) le n, forall a}$$
        By pointwise boundedness, we see that $E_n nearrow X$. Since $E_n$ are closed and $X$ is a Baire space, we see that there exists $E_n$ with an interior point $x$. Since $X$ is Frechet, its topology is generated by countable semi-norms $p_n$ and thus there exists $N,r$ such that
        $$
        x + bigcap_{k=1}^N {p_k < r} subseteq E_n
        $$
        Hence,
        $$
        bigcap_{k=1}^N {p_k < 2r} subseteq E_n -E_n subseteq E_{2n}
        $$
        Let
        $$
        p(x) = frac{1}{r} sum_{k=1}^N p_k (x)
        $$
        Hence,
        $$
        0in {p < 1} subseteq {sup {q(T_a x):ain A} <3n}
        $$
        Since $p$ is continuous, we see that $0$ is an interior point of ${sup {q(T_a x):ain A} <3n} $ and thus $sup {q(T_a x):ain A}$ is a continuous semi-norm.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 6 at 20:48

























        answered Jan 6 at 10:28









        Andrew YuanAndrew Yuan

        530210




        530210






























            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%2f2100943%2funiform-boundedness-principle-for-bounded-linear-maps-from-frechet-space-into-a%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

            Wiesbaden

            Image
            Wappen Deutschlandkarte 50.0825 8.24 117 Koordinaten: 50° 5′  N , 8° 14′  O Basisdaten Bundesland: Hessen Regierungsbezirk: Darmstadt Höhe: 117 m ü. NHN Fläche: 203,93 km 2 Einwohner: 278.654 (31. Dez. 2017) [1] Bevölkerungsdichte: 1366 Einwohner je km 2 Postleitzahlen: 65183–65207, 55246, 55252 Vorlage:Infobox Gemeinde in Deutschland/Wartung/PLZ enthält Text Vorwahlen: 0611, 06122, 06127, 06134 Kfz-Kennzeichen: WI Gemeindeschlüssel: 06 4 14 000 LOCODE: DE WIB NUTS: DE714 Stadtgliederung: 26 Ortsbezirke Adresse der Stadtverwaltung: Schlossplatz 6 65183 Wiesbaden Webpräsenz: www.wiesbaden.de Oberbürgermeister: Sven Gerich (SPD) Lage der Landeshauptstadt Wiesbaden in Hessen und im Regierungsbezirk Darmstadt Offizielles Logo der Landeshauptstadt Wiesbaden Flagge der Landeshauptstadt Wiesbaden Wiesbaden is
            Read more

            Marschland

            Image
            Kilometerweite Marsch in den Vier- und Marschlanden am Elbdeich Die Wedeler Marschlandschaft in Schleswig-Holstein Der Hadelner Kanal zur Entwässerung der Marsch Die tief gelegene Marsch und ein Entwässerungskanal Das Schöpfwerk Otterndorf im Land Hadeln zur Entwässerung der Medem verfügte damals bereits über die größte Kreiselpumpe Europas. Das Schöpfwerk in Neuhaus für die Aue und des Neuhaus-Bülkauer Kanals Als Marsch(land) (v. niederdt., altsächs. mersc ) – auch Masch , Mersch oder Schwemmland genannt – bezeichnet man eine nacheiszeitlich entstandene geomorphologische Landform im Gebiet der nordwestdeutschen Küsten und Flüsse sowie vergleichbare Landformen weltweit. Inhaltsverzeichnis 1 Landschaft 2 Entstehung 3 Marschböden 4 Nutzung 5 Marschgebiete in Deutschland 6 Siehe auch 7 Literatur 8 Weblinks 9 Einzelnachweise Landschaft | Marschen sind generell flache Landstriche ohne natürliche Erhebungen. Si
            Read more

            Dieringhausen

            Image
            Dieringhausen Stadt Gummersbach 50.982777777778 7.5316666666667 165 Koordinaten: 50° 58′ 58″  N , 7° 31′ 54″  O Höhe: 165  (160–245)  m Einwohner: 5055  (30. Jun. 2016) Postleitzahl: 51645 Vorwahl: 02261 Lage von Dieringhausen in Gummersbach Im Hohl ca. 1927–1930 Ansicht von Südwesten bei Bünghausen Dieringhausen (im örtlichen Dialekt Dierkesen ) ist ein Ortsteil von Gummersbach im Oberbergischen Kreis, Regierungsbezirk Köln, Nordrhein-Westfalen (Deutschland). Inhaltsverzeichnis 1 Geographie 2 Geschichte 3 Kultur 3.1 Sehenswürdigkeiten 3.2 Regelmäßige Veranstaltungen 4 Infrastruktur und Wirtschaft 4.1 Verkehr 4.1.1 Schienen- und Busverkehr 4.1.2 Straßen 4.2 Öffentliche Einrichtungen 4.2.1 Schulen und Bildungseinrichtungen 4.2.2 Kirchliche Einrichtungen 5 Persönlichkeiten 5.1 In Dieringhausen geboren 5.2 In Dieringhausen gelebt 6 Einzelnachweise 7
            Read more