$M={(x,y)mapsto sum_{j=1}^{n}f_j(x)g_j(y)}$ Then sum and product of such functions is again in $M$












0












$begingroup$


Let $M$ be the set of all functions of the type $(x,y)mapsto sum_{j=1}^{n}f_j(x)g_j(y)$ with $f_j in C(X, mathbb{R})$ and $g_j in C(Y, mathbb{R})$ where $X,Y$ are compact topological spaces.



I need to show that the sum and product of two of such functions $a:=sum_{j=1}^{n}f^a_j(x)g^a_j(y)$ and $b:=sum_{i=1}^{m}f^b_i(x)g^b_i(y)$ is again in $M$. Note that $^a$ and $^b$ it just for notational reasons, so $f^a$ doens't mean $f$ multiplied by itself $a$ times



For the product:



$ab=sum_{j=1}^{n}f^a_j(x)g^a_j(y)sum_{i=1}^{m}f^b_i(x)g^b_i(y)=sum_{(j,i) in {1ldots n }times{1ldots m}}f^a_jf_i^b(x)g_j^ag_i^b(y)$
Does this already show that $ab$ is in $M$?



For the sum:



$a+b=sum_{j=1}^{n}f^a_j(x)g^a_j(y)+sum_{i=1}^{m}f^b_i(x)g^b_i(y)$. But how can I write this as one sum?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    it already is written as a sum, from 1 to (m+n) of products of functions of x times functions of y
    $endgroup$
    – Sorin Tirc
    Dec 20 '18 at 14:49
















0












$begingroup$


Let $M$ be the set of all functions of the type $(x,y)mapsto sum_{j=1}^{n}f_j(x)g_j(y)$ with $f_j in C(X, mathbb{R})$ and $g_j in C(Y, mathbb{R})$ where $X,Y$ are compact topological spaces.



I need to show that the sum and product of two of such functions $a:=sum_{j=1}^{n}f^a_j(x)g^a_j(y)$ and $b:=sum_{i=1}^{m}f^b_i(x)g^b_i(y)$ is again in $M$. Note that $^a$ and $^b$ it just for notational reasons, so $f^a$ doens't mean $f$ multiplied by itself $a$ times



For the product:



$ab=sum_{j=1}^{n}f^a_j(x)g^a_j(y)sum_{i=1}^{m}f^b_i(x)g^b_i(y)=sum_{(j,i) in {1ldots n }times{1ldots m}}f^a_jf_i^b(x)g_j^ag_i^b(y)$
Does this already show that $ab$ is in $M$?



For the sum:



$a+b=sum_{j=1}^{n}f^a_j(x)g^a_j(y)+sum_{i=1}^{m}f^b_i(x)g^b_i(y)$. But how can I write this as one sum?










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    it already is written as a sum, from 1 to (m+n) of products of functions of x times functions of y
    $endgroup$
    – Sorin Tirc
    Dec 20 '18 at 14:49














0












0








0





$begingroup$


Let $M$ be the set of all functions of the type $(x,y)mapsto sum_{j=1}^{n}f_j(x)g_j(y)$ with $f_j in C(X, mathbb{R})$ and $g_j in C(Y, mathbb{R})$ where $X,Y$ are compact topological spaces.



I need to show that the sum and product of two of such functions $a:=sum_{j=1}^{n}f^a_j(x)g^a_j(y)$ and $b:=sum_{i=1}^{m}f^b_i(x)g^b_i(y)$ is again in $M$. Note that $^a$ and $^b$ it just for notational reasons, so $f^a$ doens't mean $f$ multiplied by itself $a$ times



For the product:



$ab=sum_{j=1}^{n}f^a_j(x)g^a_j(y)sum_{i=1}^{m}f^b_i(x)g^b_i(y)=sum_{(j,i) in {1ldots n }times{1ldots m}}f^a_jf_i^b(x)g_j^ag_i^b(y)$
Does this already show that $ab$ is in $M$?



For the sum:



$a+b=sum_{j=1}^{n}f^a_j(x)g^a_j(y)+sum_{i=1}^{m}f^b_i(x)g^b_i(y)$. But how can I write this as one sum?










share|cite|improve this question









$endgroup$




Let $M$ be the set of all functions of the type $(x,y)mapsto sum_{j=1}^{n}f_j(x)g_j(y)$ with $f_j in C(X, mathbb{R})$ and $g_j in C(Y, mathbb{R})$ where $X,Y$ are compact topological spaces.



I need to show that the sum and product of two of such functions $a:=sum_{j=1}^{n}f^a_j(x)g^a_j(y)$ and $b:=sum_{i=1}^{m}f^b_i(x)g^b_i(y)$ is again in $M$. Note that $^a$ and $^b$ it just for notational reasons, so $f^a$ doens't mean $f$ multiplied by itself $a$ times



For the product:



$ab=sum_{j=1}^{n}f^a_j(x)g^a_j(y)sum_{i=1}^{m}f^b_i(x)g^b_i(y)=sum_{(j,i) in {1ldots n }times{1ldots m}}f^a_jf_i^b(x)g_j^ag_i^b(y)$
Does this already show that $ab$ is in $M$?



For the sum:



$a+b=sum_{j=1}^{n}f^a_j(x)g^a_j(y)+sum_{i=1}^{m}f^b_i(x)g^b_i(y)$. But how can I write this as one sum?







real-analysis calculus






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 20 '18 at 13:42









user626880user626880

204




204








  • 1




    $begingroup$
    it already is written as a sum, from 1 to (m+n) of products of functions of x times functions of y
    $endgroup$
    – Sorin Tirc
    Dec 20 '18 at 14:49














  • 1




    $begingroup$
    it already is written as a sum, from 1 to (m+n) of products of functions of x times functions of y
    $endgroup$
    – Sorin Tirc
    Dec 20 '18 at 14:49








1




1




$begingroup$
it already is written as a sum, from 1 to (m+n) of products of functions of x times functions of y
$endgroup$
– Sorin Tirc
Dec 20 '18 at 14:49




$begingroup$
it already is written as a sum, from 1 to (m+n) of products of functions of x times functions of y
$endgroup$
– Sorin Tirc
Dec 20 '18 at 14:49










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%2f3047551%2fm-x-y-mapsto-sum-j-1nf-jxg-jy-then-sum-and-product-of-such-fun%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%2f3047551%2fm-x-y-mapsto-sum-j-1nf-jxg-jy-then-sum-and-product-of-such-fun%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

Quarter-circle Tiles

build a pushdown automaton that recognizes the reverse language of a given pushdown automaton?

Mont Emei