Converse of the projection theorem
I am trying to prove the converse of the projection theorem:
If for every $fin H$ there is a $pin M$ such that $|p−f|=inflimits_{vin M}|v−f|$, then $M$ is closed.
Is my proof correct?
Let $p_nin M,n=1,2,ldots,$ be a sequence converging to $gin H$. Then there is a $pin M$ such that $|p−g|=inflimits_{vin M}|v−g|$, so $|p−g|leq |p_n-g|,$ for all $n$. Now as $n$ goes to infinity the RHS goes to zero, so $g=pin M$. Therefore, $M$ contains all its limit points and is closed.
functional-analysis analysis hilbert-spaces
edited Dec 4 '18 at 15:05
Hanno
2,061425
asked Nov 8 '17 at 9:05
RozaTh
190210
add a comment |
I am trying to prove the converse of the projection theorem:
If for every $fin H$ there is a $pin M$ such that $|p−f|=inflimits_{vin M}|v−f|$, then $M$ is closed.
Is my proof correct?
Let $p_nin M,n=1,2,ldots,$ be a sequence converging to $gin H$. Then there is a $pin M$ such that $|p−g|=inflimits_{vin M}|v−g|$, so $|p−g|leq |p_n-g|,$ for all $n$. Now as $n$ goes to infinity the RHS goes to zero, so $g=pin M$. Therefore, $M$ contains all its limit points and is closed.
functional-analysis analysis hilbert-spaces
edited Dec 4 '18 at 15:05
Hanno
2,061425
asked Nov 8 '17 at 9:05
RozaTh
190210
3
Your argument is correct (even in more general metric spaces instead of Hilbert space).
– Jochen
Nov 8 '17 at 9:25
add a comment |
I am trying to prove the converse of the projection theorem:
If for every $fin H$ there is a $pin M$ such that $|p−f|=inflimits_{vin M}|v−f|$, then $M$ is closed.
Is my proof correct?
Let $p_nin M,n=1,2,ldots,$ be a sequence converging to $gin H$. Then there is a $pin M$ such that $|p−g|=inflimits_{vin M}|v−g|$, so $|p−g|leq |p_n-g|,$ for all $n$. Now as $n$ goes to infinity the RHS goes to zero, so $g=pin M$. Therefore, $M$ contains all its limit points and is closed.
functional-analysis analysis hilbert-spaces
edited Dec 4 '18 at 15:05
Hanno
2,061425
asked Nov 8 '17 at 9:05
RozaTh
190210
I am trying to prove the converse of the projection theorem:
If for every $fin H$ there is a $pin M$ such that $|p−f|=inflimits_{vin M}|v−f|$, then $M$ is closed.
Is my proof correct?
Let $p_nin M,n=1,2,ldots,$ be a sequence converging to $gin H$. Then there is a $pin M$ such that $|p−g|=inflimits_{vin M}|v−g|$, so $|p−g|leq |p_n-g|,$ for all $n$. Now as $n$ goes to infinity the RHS goes to zero, so $g=pin M$. Therefore, $M$ contains all its limit points and is closed.
functional-analysis analysis hilbert-spaces
functional-analysis analysis hilbert-spaces
edited Dec 4 '18 at 15:05
Hanno
2,061425
asked Nov 8 '17 at 9:05
RozaTh
190210
edited Dec 4 '18 at 15:05
Hanno
2,061425
asked Nov 8 '17 at 9:05
RozaTh
190210
edited Dec 4 '18 at 15:05
Hanno
2,061425
edited Dec 4 '18 at 15:05
Hanno
2,061425
edited Dec 4 '18 at 15:05
Hanno
2,061425
2,061425
asked Nov 8 '17 at 9:05
RozaTh
190210
asked Nov 8 '17 at 9:05
RozaTh
190210
asked Nov 8 '17 at 9:05
RozaTh
190210
190210
3
Your argument is correct (even in more general metric spaces instead of Hilbert space).
– Jochen
Nov 8 '17 at 9:25
add a comment |
3
Your argument is correct (even in more general metric spaces instead of Hilbert space).
– Jochen
Nov 8 '17 at 9:25
3
3
Your argument is correct (even in more general metric spaces instead of Hilbert space).
– Jochen
Nov 8 '17 at 9:25
Your argument is correct (even in more general metric spaces instead of Hilbert space).
– Jochen
Nov 8 '17 at 9:25
add a comment |
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
});
}
});
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%2f2510381%2fconverse-of-the-projection-theorem%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
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f2510381%2fconverse-of-the-projection-theorem%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
3
Your argument is correct (even in more general metric spaces instead of Hilbert space).
– Jochen
Nov 8 '17 at 9:25