Prove that $S_{n} : X to X$ is continuous mapping.
$begingroup$
Consider $X$ - Banach space with Schauder basis : $forall x in X$ $exists!$ $x = sum_{i=1}^{infty} x_{i}e_{i}$.
Let's consider $S_{n} = sum_{i = 1}^{n} x_{i}e_{i}$. We want to show that it is a continous mapping (i.e. $exists C > 0: $ $|S_n| le C |x|$, for all $x in X$).
How should I start with? Should I use Cauchy-sequences?
real-analysis functional-analysis banach-spaces
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
$endgroup$
add a comment |
$begingroup$
Consider $X$ - Banach space with Schauder basis : $forall x in X$ $exists!$ $x = sum_{i=1}^{infty} x_{i}e_{i}$.
Let's consider $S_{n} = sum_{i = 1}^{n} x_{i}e_{i}$. We want to show that it is a continous mapping (i.e. $exists C > 0: $ $|S_n| le C |x|$, for all $x in X$).
How should I start with? Should I use Cauchy-sequences?
real-analysis functional-analysis banach-spaces
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
$endgroup$
add a comment |
$begingroup$
Consider $X$ - Banach space with Schauder basis : $forall x in X$ $exists!$ $x = sum_{i=1}^{infty} x_{i}e_{i}$.
Let's consider $S_{n} = sum_{i = 1}^{n} x_{i}e_{i}$. We want to show that it is a continous mapping (i.e. $exists C > 0: $ $|S_n| le C |x|$, for all $x in X$).
How should I start with? Should I use Cauchy-sequences?
real-analysis functional-analysis banach-spaces
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
$endgroup$
Consider $X$ - Banach space with Schauder basis : $forall x in X$ $exists!$ $x = sum_{i=1}^{infty} x_{i}e_{i}$.
Let's consider $S_{n} = sum_{i = 1}^{n} x_{i}e_{i}$. We want to show that it is a continous mapping (i.e. $exists C > 0: $ $|S_n| le C |x|$, for all $x in X$).
How should I start with? Should I use Cauchy-sequences?
real-analysis functional-analysis banach-spaces
real-analysis functional-analysis banach-spaces
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
asked Dec 20 '18 at 17:45
openspaceopenspace
3,4512822
3,4512822
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
$x = sum_{i=1}^infty x_i e_i$ means that the partial sums converge to $x$ in norm. So, if you add and subtract $x$ inside $|S_n|$, then apply the triangle inequality, you should be able to finish the proof.
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
$endgroup$
add a comment |
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
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3047797%2fprove-that-s-n-x-to-x-is-continuous-mapping%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
$begingroup$
$x = sum_{i=1}^infty x_i e_i$ means that the partial sums converge to $x$ in norm. So, if you add and subtract $x$ inside $|S_n|$, then apply the triangle inequality, you should be able to finish the proof.
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
$endgroup$
add a comment |
$begingroup$
$x = sum_{i=1}^infty x_i e_i$ means that the partial sums converge to $x$ in norm. So, if you add and subtract $x$ inside $|S_n|$, then apply the triangle inequality, you should be able to finish the proof.
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
$endgroup$
add a comment |
$begingroup$
$x = sum_{i=1}^infty x_i e_i$ means that the partial sums converge to $x$ in norm. So, if you add and subtract $x$ inside $|S_n|$, then apply the triangle inequality, you should be able to finish the proof.
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
$endgroup$
$x = sum_{i=1}^infty x_i e_i$ means that the partial sums converge to $x$ in norm. So, if you add and subtract $x$ inside $|S_n|$, then apply the triangle inequality, you should be able to finish the proof.
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
answered Dec 20 '18 at 19:05
DunhamDunham
2,074614
2,074614
add a comment |
add a comment |
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.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3047797%2fprove-that-s-n-x-to-x-is-continuous-mapping%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
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
