Task concerning Stone Weierstrass












0












$begingroup$


Let $D$ be the open unit circle with center $0$ in $mathbb{C}$.



Let $overline D$ be the closed unit circle with center $0$ in $mathbb{C}$.



Let $mathbb{C}[z]_{big|overline D}$ be the set of all functions $f:overline Dtomathbb{C}$ of polynomials of the type $f(z)=sum_{j=0}^{N}a_jz^j$.



The task has something to do with the complex version of Stone-Weierstrass.




I need to show which of the conditions for the sentence are not fulfilled:




$mathbb{C}[z]_{big|overline D}$ is a algebra in $C_b(D,mathbb{C})$ where $C_b(D,mathbb{C})$ describes the set of all continuous and bounded functions from $D$ to $mathbb{C}$. I think there should be no problem because $mathbb{C}[z]_{big|overline D}$ is a linear subspace and for $f,g in mathbb{C}[z]_{big|overline D}$ $Rightarrow fg in mathbb{C}[z]_{big|overline D}$.



Then the next one: $z_1 neq z_2 Rightarrow f(z_1) neq f(z_2)$ should be no problem either by choosing $f(z)=z$ And for every $z$ there exists a $f$ such that $f(z)neq 0$ by simply choosing $f=1$ constant.



So If I didn't make any mistakes so far the only thing that can go wrong is $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$.



$overline f(z)=sum_{j=0}^{N}overline a_j z^{-j}$. Is this not in $mathbb{C}[z]_{big|overline D}$ because $z$ can have negative exponents?






real-analysis complex-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    It is true that $sum_{j=0}^N overline a_j z^{-j}$ is not in $left.Bbb C[z]rightrvert_{overline D}$, but it's not true that $overline f(z)=sum_{j=0}^N overline a_j z^{-j}$ (unless $N=0$). In fact, $$overline f(z)=sum_{j=0}^N overline a_j overline z^{j}.$$
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 16:38










  • $begingroup$
    So for $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$ is fulfilled?
    $endgroup$
    – conrad
    Dec 15 '18 at 16:51










  • $begingroup$
    I'm wondering, is $overline f$ a polynomial of the type $f(z)=sum_{j=0}^{N}a_jz^j$?
    $endgroup$
    – conrad
    Dec 15 '18 at 17:00












  • $begingroup$
    No, it isn't${}$.
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 21:20
















0












$begingroup$


Let $D$ be the open unit circle with center $0$ in $mathbb{C}$.



Let $overline D$ be the closed unit circle with center $0$ in $mathbb{C}$.



Let $mathbb{C}[z]_{big|overline D}$ be the set of all functions $f:overline Dtomathbb{C}$ of polynomials of the type $f(z)=sum_{j=0}^{N}a_jz^j$.



The task has something to do with the complex version of Stone-Weierstrass.




I need to show which of the conditions for the sentence are not fulfilled:




$mathbb{C}[z]_{big|overline D}$ is a algebra in $C_b(D,mathbb{C})$ where $C_b(D,mathbb{C})$ describes the set of all continuous and bounded functions from $D$ to $mathbb{C}$. I think there should be no problem because $mathbb{C}[z]_{big|overline D}$ is a linear subspace and for $f,g in mathbb{C}[z]_{big|overline D}$ $Rightarrow fg in mathbb{C}[z]_{big|overline D}$.



Then the next one: $z_1 neq z_2 Rightarrow f(z_1) neq f(z_2)$ should be no problem either by choosing $f(z)=z$ And for every $z$ there exists a $f$ such that $f(z)neq 0$ by simply choosing $f=1$ constant.



So If I didn't make any mistakes so far the only thing that can go wrong is $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$.



$overline f(z)=sum_{j=0}^{N}overline a_j z^{-j}$. Is this not in $mathbb{C}[z]_{big|overline D}$ because $z$ can have negative exponents?






real-analysis complex-analysis







share|cite|improve this question











$endgroup$












  • $begingroup$
    It is true that $sum_{j=0}^N overline a_j z^{-j}$ is not in $left.Bbb C[z]rightrvert_{overline D}$, but it's not true that $overline f(z)=sum_{j=0}^N overline a_j z^{-j}$ (unless $N=0$). In fact, $$overline f(z)=sum_{j=0}^N overline a_j overline z^{j}.$$
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 16:38










  • $begingroup$
    So for $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$ is fulfilled?
    $endgroup$
    – conrad
    Dec 15 '18 at 16:51










  • $begingroup$
    I'm wondering, is $overline f$ a polynomial of the type $f(z)=sum_{j=0}^{N}a_jz^j$?
    $endgroup$
    – conrad
    Dec 15 '18 at 17:00












  • $begingroup$
    No, it isn't${}$.
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 21:20














0












0








0





$begingroup$


Let $D$ be the open unit circle with center $0$ in $mathbb{C}$.



Let $overline D$ be the closed unit circle with center $0$ in $mathbb{C}$.



Let $mathbb{C}[z]_{big|overline D}$ be the set of all functions $f:overline Dtomathbb{C}$ of polynomials of the type $f(z)=sum_{j=0}^{N}a_jz^j$.



The task has something to do with the complex version of Stone-Weierstrass.




I need to show which of the conditions for the sentence are not fulfilled:




$mathbb{C}[z]_{big|overline D}$ is a algebra in $C_b(D,mathbb{C})$ where $C_b(D,mathbb{C})$ describes the set of all continuous and bounded functions from $D$ to $mathbb{C}$. I think there should be no problem because $mathbb{C}[z]_{big|overline D}$ is a linear subspace and for $f,g in mathbb{C}[z]_{big|overline D}$ $Rightarrow fg in mathbb{C}[z]_{big|overline D}$.



Then the next one: $z_1 neq z_2 Rightarrow f(z_1) neq f(z_2)$ should be no problem either by choosing $f(z)=z$ And for every $z$ there exists a $f$ such that $f(z)neq 0$ by simply choosing $f=1$ constant.



So If I didn't make any mistakes so far the only thing that can go wrong is $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$.



$overline f(z)=sum_{j=0}^{N}overline a_j z^{-j}$. Is this not in $mathbb{C}[z]_{big|overline D}$ because $z$ can have negative exponents?






real-analysis complex-analysis







share|cite|improve this question











$endgroup$




Let $D$ be the open unit circle with center $0$ in $mathbb{C}$.



Let $overline D$ be the closed unit circle with center $0$ in $mathbb{C}$.



Let $mathbb{C}[z]_{big|overline D}$ be the set of all functions $f:overline Dtomathbb{C}$ of polynomials of the type $f(z)=sum_{j=0}^{N}a_jz^j$.



The task has something to do with the complex version of Stone-Weierstrass.




I need to show which of the conditions for the sentence are not fulfilled:




$mathbb{C}[z]_{big|overline D}$ is a algebra in $C_b(D,mathbb{C})$ where $C_b(D,mathbb{C})$ describes the set of all continuous and bounded functions from $D$ to $mathbb{C}$. I think there should be no problem because $mathbb{C}[z]_{big|overline D}$ is a linear subspace and for $f,g in mathbb{C}[z]_{big|overline D}$ $Rightarrow fg in mathbb{C}[z]_{big|overline D}$.



Then the next one: $z_1 neq z_2 Rightarrow f(z_1) neq f(z_2)$ should be no problem either by choosing $f(z)=z$ And for every $z$ there exists a $f$ such that $f(z)neq 0$ by simply choosing $f=1$ constant.



So If I didn't make any mistakes so far the only thing that can go wrong is $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$.



$overline f(z)=sum_{j=0}^{N}overline a_j z^{-j}$. Is this not in $mathbb{C}[z]_{big|overline D}$ because $z$ can have negative exponents?






real-analysis complex-analysis




real-analysis complex-analysis






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 15 '18 at 17:00







conrad

















asked Dec 15 '18 at 16:34









conradconrad

757




757












  • $begingroup$
    It is true that $sum_{j=0}^N overline a_j z^{-j}$ is not in $left.Bbb C[z]rightrvert_{overline D}$, but it's not true that $overline f(z)=sum_{j=0}^N overline a_j z^{-j}$ (unless $N=0$). In fact, $$overline f(z)=sum_{j=0}^N overline a_j overline z^{j}.$$
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 16:38










  • $begingroup$
    So for $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$ is fulfilled?
    $endgroup$
    – conrad
    Dec 15 '18 at 16:51










  • $begingroup$
    I'm wondering, is $overline f$ a polynomial of the type $f(z)=sum_{j=0}^{N}a_jz^j$?
    $endgroup$
    – conrad
    Dec 15 '18 at 17:00












  • $begingroup$
    No, it isn't${}$.
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 21:20


















  • $begingroup$
    It is true that $sum_{j=0}^N overline a_j z^{-j}$ is not in $left.Bbb C[z]rightrvert_{overline D}$, but it's not true that $overline f(z)=sum_{j=0}^N overline a_j z^{-j}$ (unless $N=0$). In fact, $$overline f(z)=sum_{j=0}^N overline a_j overline z^{j}.$$
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 16:38










  • $begingroup$
    So for $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$ is fulfilled?
    $endgroup$
    – conrad
    Dec 15 '18 at 16:51










  • $begingroup$
    I'm wondering, is $overline f$ a polynomial of the type $f(z)=sum_{j=0}^{N}a_jz^j$?
    $endgroup$
    – conrad
    Dec 15 '18 at 17:00












  • $begingroup$
    No, it isn't${}$.
    $endgroup$
    – Saucy O'Path
    Dec 15 '18 at 21:20
















$begingroup$
It is true that $sum_{j=0}^N overline a_j z^{-j}$ is not in $left.Bbb C[z]rightrvert_{overline D}$, but it's not true that $overline f(z)=sum_{j=0}^N overline a_j z^{-j}$ (unless $N=0$). In fact, $$overline f(z)=sum_{j=0}^N overline a_j overline z^{j}.$$
$endgroup$
– Saucy O'Path
Dec 15 '18 at 16:38




$begingroup$
It is true that $sum_{j=0}^N overline a_j z^{-j}$ is not in $left.Bbb C[z]rightrvert_{overline D}$, but it's not true that $overline f(z)=sum_{j=0}^N overline a_j z^{-j}$ (unless $N=0$). In fact, $$overline f(z)=sum_{j=0}^N overline a_j overline z^{j}.$$
$endgroup$
– Saucy O'Path
Dec 15 '18 at 16:38












$begingroup$
So for $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$ is fulfilled?
$endgroup$
– conrad
Dec 15 '18 at 16:51




$begingroup$
So for $f in mathbb{C}[z]_{big|overline D} Rightarrow overline f in mathbb{C}[z]_{big|overline D}$ is fulfilled?
$endgroup$
– conrad
Dec 15 '18 at 16:51












$begingroup$
I'm wondering, is $overline f$ a polynomial of the type $f(z)=sum_{j=0}^{N}a_jz^j$?
$endgroup$
– conrad
Dec 15 '18 at 17:00






$begingroup$
I'm wondering, is $overline f$ a polynomial of the type $f(z)=sum_{j=0}^{N}a_jz^j$?
$endgroup$
– conrad
Dec 15 '18 at 17:00














$begingroup$
No, it isn't${}$.
$endgroup$
– Saucy O'Path
Dec 15 '18 at 21:20




$begingroup$
No, it isn't${}$.
$endgroup$
– Saucy O'Path
Dec 15 '18 at 21:20










0






active

oldest

votes











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%2f3041684%2ftask-concerning-stone-weierstrass%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























0






active

oldest

votes








0






active

oldest

votes









active

oldest

votes






active

oldest

votes
















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%2f3041684%2ftask-concerning-stone-weierstrass%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