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

Probability when a professor distributes a quiz and homework assignment to a class of n students.

Image
0 0 $begingroup$ Need help with this problem. Suppose our lazy professor collects a quiz and a homework assignment from a class of n students one day, then distributes both the quizzes and the homework assignments back to the class in a random fashion for grading. Each student receives one quiz and one homework assignment to grade. (a) What is the probability that every student receives someone else's quiz to grade, and someone else's homework to grade? (b) What is the probability that no student receives both their own quiz and their own homework assignment to grade? In this case, some students may receive their own quiz, and others may receive their own homework assignment. (c) Compute the limiting probability as n approaches infinity in each case. ...
Read more

Aardman Animations

Image
Aardman Animations, Ltd. Aardman Animations Tipo privada Atividade Animação em stop-motion , Animação em CGI Fundação 1972 Fundador(es) Peter Lord David Sproxton Sede Bristol, Inglaterra  Reino Unido Proprietário(s) Dreamworks Pessoas-chave Peter Lord David Sproxton Nick Park Divisões Aardman Features Aardman Digital Aardman Commercials Aardman Broadcast Aardman International Aardman Rights Aardman Effects Aardman 3-D Systems Aardman Nathan Love AardBoiled Website oficial aardman.com ...
Read more

Are they similar matrix

Image
0 $begingroup$ Do $begin{bmatrix} 0&i&0\0&0&1\0&0&0 end{bmatrix} $ and $begin{bmatrix} 0&0&0\-i&0&0\0&1&0 end{bmatrix} $ are similar.Is this True/false Clearly both are nilpotent and one is conjucate transpose of other but how to know if they are similar.i'm stuck. Please help me linear-algebra vector-spaces eigenvalues-eigenvectors generalizedeigenvector share | cite | improve this question edited Dec 10 '18 at 6:38 Vasanth Kris asked Dec 10 '18 at 6:19 ...
Read more