Weak convergence in closed subspace of a Hilbert Space
I get stuck in following problems
Let $(x_n)_n subset Y$ be such that $x_n xrightarrow{w} x$ (converges weakly), and Y a closed subspace of a Hilbert Space. Show that $x in Y$
My try
I didn't make a great progress.
I just know that
For every $y in H$ we have that $<x_n, y> rightarrow <x, y>$ iff $|<x_n, y> - <x, y>| rightarrow 0$.
I think that I have to show a subsequence of $(x_n)$ which strongly converges to $x$. (I get stuck here)
Also I've to show
Let $T in B(H)$ and $x_n xrightarrow{w} x$ (converges weakly) then $Tx_n xrightarrow{w} Tx$ (converges weakly).
I think that I've to solve the previous problem before this.
Thank you.
functional-analysis hilbert-spaces weak-convergence
asked Nov 27 '18 at 20:00
Kutz
327
add a comment |
I get stuck in following problems
Let $(x_n)_n subset Y$ be such that $x_n xrightarrow{w} x$ (converges weakly), and Y a closed subspace of a Hilbert Space. Show that $x in Y$
My try
I didn't make a great progress.
I just know that
For every $y in H$ we have that $<x_n, y> rightarrow <x, y>$ iff $|<x_n, y> - <x, y>| rightarrow 0$.
I think that I have to show a subsequence of $(x_n)$ which strongly converges to $x$. (I get stuck here)
Also I've to show
Let $T in B(H)$ and $x_n xrightarrow{w} x$ (converges weakly) then $Tx_n xrightarrow{w} Tx$ (converges weakly).
I think that I've to solve the previous problem before this.
Thank you.
functional-analysis hilbert-spaces weak-convergence
asked Nov 27 '18 at 20:00
Kutz
327
add a comment |
I get stuck in following problems
Let $(x_n)_n subset Y$ be such that $x_n xrightarrow{w} x$ (converges weakly), and Y a closed subspace of a Hilbert Space. Show that $x in Y$
My try
I didn't make a great progress.
I just know that
For every $y in H$ we have that $<x_n, y> rightarrow <x, y>$ iff $|<x_n, y> - <x, y>| rightarrow 0$.
I think that I have to show a subsequence of $(x_n)$ which strongly converges to $x$. (I get stuck here)
Also I've to show
Let $T in B(H)$ and $x_n xrightarrow{w} x$ (converges weakly) then $Tx_n xrightarrow{w} Tx$ (converges weakly).
I think that I've to solve the previous problem before this.
Thank you.
functional-analysis hilbert-spaces weak-convergence
asked Nov 27 '18 at 20:00
Kutz
327
I get stuck in following problems
Let $(x_n)_n subset Y$ be such that $x_n xrightarrow{w} x$ (converges weakly), and Y a closed subspace of a Hilbert Space. Show that $x in Y$
My try
I didn't make a great progress.
I just know that
For every $y in H$ we have that $<x_n, y> rightarrow <x, y>$ iff $|<x_n, y> - <x, y>| rightarrow 0$.
I think that I have to show a subsequence of $(x_n)$ which strongly converges to $x$. (I get stuck here)
Also I've to show
Let $T in B(H)$ and $x_n xrightarrow{w} x$ (converges weakly) then $Tx_n xrightarrow{w} Tx$ (converges weakly).
I think that I've to solve the previous problem before this.
Thank you.
functional-analysis hilbert-spaces weak-convergence
functional-analysis hilbert-spaces weak-convergence
asked Nov 27 '18 at 20:00
Kutz
327
asked Nov 27 '18 at 20:00
Kutz
327
asked Nov 27 '18 at 20:00
Kutz
327
asked Nov 27 '18 at 20:00
Kutz
327
asked Nov 27 '18 at 20:00
Kutz
327
327
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
If $Y$ is closed, $Y={cap_{fin H^*}Kerf, Ysubset Ker f}$. To see this, remark that $Ysubset {cap_{fin H^*}Kerf, Ysubset Ker f}$. Let $yin {cap_{fin H^*}Kerf, Ysubset Ker f}$, suppose that $y$ is not in $Y$, consider $f$ defined on $Z=Vect(Y,y)$ such that $f(y)=1, f(Y)=0$, $f$ is bounded on $Z$. By Hahn Banach you can extend it to $H$. Contradiction.
Suppose that $fin H^*$ and $Ysubset Ker f$, $f(x_n)=0$ implies that $f(x)=0$, implies that $xin {cap_{fin H^*}Kerf, Ysubset Ker f}=Y$.
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
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%2f3016227%2fweak-convergence-in-closed-subspace-of-a-hilbert-space%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
If $Y$ is closed, $Y={cap_{fin H^*}Kerf, Ysubset Ker f}$. To see this, remark that $Ysubset {cap_{fin H^*}Kerf, Ysubset Ker f}$. Let $yin {cap_{fin H^*}Kerf, Ysubset Ker f}$, suppose that $y$ is not in $Y$, consider $f$ defined on $Z=Vect(Y,y)$ such that $f(y)=1, f(Y)=0$, $f$ is bounded on $Z$. By Hahn Banach you can extend it to $H$. Contradiction.
Suppose that $fin H^*$ and $Ysubset Ker f$, $f(x_n)=0$ implies that $f(x)=0$, implies that $xin {cap_{fin H^*}Kerf, Ysubset Ker f}=Y$.
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
add a comment |
If $Y$ is closed, $Y={cap_{fin H^*}Kerf, Ysubset Ker f}$. To see this, remark that $Ysubset {cap_{fin H^*}Kerf, Ysubset Ker f}$. Let $yin {cap_{fin H^*}Kerf, Ysubset Ker f}$, suppose that $y$ is not in $Y$, consider $f$ defined on $Z=Vect(Y,y)$ such that $f(y)=1, f(Y)=0$, $f$ is bounded on $Z$. By Hahn Banach you can extend it to $H$. Contradiction.
Suppose that $fin H^*$ and $Ysubset Ker f$, $f(x_n)=0$ implies that $f(x)=0$, implies that $xin {cap_{fin H^*}Kerf, Ysubset Ker f}=Y$.
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
add a comment |
If $Y$ is closed, $Y={cap_{fin H^*}Kerf, Ysubset Ker f}$. To see this, remark that $Ysubset {cap_{fin H^*}Kerf, Ysubset Ker f}$. Let $yin {cap_{fin H^*}Kerf, Ysubset Ker f}$, suppose that $y$ is not in $Y$, consider $f$ defined on $Z=Vect(Y,y)$ such that $f(y)=1, f(Y)=0$, $f$ is bounded on $Z$. By Hahn Banach you can extend it to $H$. Contradiction.
Suppose that $fin H^*$ and $Ysubset Ker f$, $f(x_n)=0$ implies that $f(x)=0$, implies that $xin {cap_{fin H^*}Kerf, Ysubset Ker f}=Y$.
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
If $Y$ is closed, $Y={cap_{fin H^*}Kerf, Ysubset Ker f}$. To see this, remark that $Ysubset {cap_{fin H^*}Kerf, Ysubset Ker f}$. Let $yin {cap_{fin H^*}Kerf, Ysubset Ker f}$, suppose that $y$ is not in $Y$, consider $f$ defined on $Z=Vect(Y,y)$ such that $f(y)=1, f(Y)=0$, $f$ is bounded on $Z$. By Hahn Banach you can extend it to $H$. Contradiction.
Suppose that $fin H^*$ and $Ysubset Ker f$, $f(x_n)=0$ implies that $f(x)=0$, implies that $xin {cap_{fin H^*}Kerf, Ysubset Ker f}=Y$.
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
edited Nov 27 '18 at 20:08
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
answered Nov 27 '18 at 20:03
Tsemo Aristide
56.1k11444
56.1k11444
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.
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%2f3016227%2fweak-convergence-in-closed-subspace-of-a-hilbert-space%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
