Proving that if the dual space $X^*$ of a Banach space $X$ is separable , then $X$ is separable
$begingroup$
I have reading an answer post before , and there's something I do not understand .
For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n in X$ with $|x_n | le1$ such that $f_n(x_n) ge frac{1}{2}$ .
I can show that for each $f_n$ , there exist $x in X$ , $f_n(x)=a neq 0$ . So let $x_n = frac{x}{a}$ , we get the inequality . But how to get the desired $x_n$ with $x_n$ also in the unit ball of $X$ ?
functional-analysis banach-spaces dual-spaces separable-spaces
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
$endgroup$
add a comment |
$begingroup$
I have reading an answer post before , and there's something I do not understand .
For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n in X$ with $|x_n | le1$ such that $f_n(x_n) ge frac{1}{2}$ .
I can show that for each $f_n$ , there exist $x in X$ , $f_n(x)=a neq 0$ . So let $x_n = frac{x}{a}$ , we get the inequality . But how to get the desired $x_n$ with $x_n$ also in the unit ball of $X$ ?
functional-analysis banach-spaces dual-spaces separable-spaces
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
$endgroup$
1
$begingroup$
I think that both comments and the answer there explain that the proof as written there assumes that $f_n$ belong to the unit sphere, i.e., $|f_n|=1$. Using that, you can get $x_n$ such that $f_n(x_n)gefrac12$.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 13:43
add a comment |
$begingroup$
I have reading an answer post before , and there's something I do not understand .
For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n in X$ with $|x_n | le1$ such that $f_n(x_n) ge frac{1}{2}$ .
I can show that for each $f_n$ , there exist $x in X$ , $f_n(x)=a neq 0$ . So let $x_n = frac{x}{a}$ , we get the inequality . But how to get the desired $x_n$ with $x_n$ also in the unit ball of $X$ ?
functional-analysis banach-spaces dual-spaces separable-spaces
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
$endgroup$
I have reading an answer post before , and there's something I do not understand .
For each $f_n$ in the unit ball of $X^*$ , why there exist $x_n in X$ with $|x_n | le1$ such that $f_n(x_n) ge frac{1}{2}$ .
I can show that for each $f_n$ , there exist $x in X$ , $f_n(x)=a neq 0$ . So let $x_n = frac{x}{a}$ , we get the inequality . But how to get the desired $x_n$ with $x_n$ also in the unit ball of $X$ ?
functional-analysis banach-spaces dual-spaces separable-spaces
functional-analysis banach-spaces dual-spaces separable-spaces
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
edited Dec 8 '18 at 13:39
J.Guo
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
asked Dec 8 '18 at 13:33
J.GuoJ.Guo
3209
3209
1
$begingroup$
I think that both comments and the answer there explain that the proof as written there assumes that $f_n$ belong to the unit sphere, i.e., $|f_n|=1$. Using that, you can get $x_n$ such that $f_n(x_n)gefrac12$.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 13:43
add a comment |
1
$begingroup$
I think that both comments and the answer there explain that the proof as written there assumes that $f_n$ belong to the unit sphere, i.e., $|f_n|=1$. Using that, you can get $x_n$ such that $f_n(x_n)gefrac12$.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 13:43
1
1
$begingroup$
I think that both comments and the answer there explain that the proof as written there assumes that $f_n$ belong to the unit sphere, i.e., $|f_n|=1$. Using that, you can get $x_n$ such that $f_n(x_n)gefrac12$.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 13:43
$begingroup$
I think that both comments and the answer there explain that the proof as written there assumes that $f_n$ belong to the unit sphere, i.e., $|f_n|=1$. Using that, you can get $x_n$ such that $f_n(x_n)gefrac12$.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 13:43
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
That is false. It should read, the unit sphere, not the unit ball.
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
$endgroup$
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
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%2f3031116%2fproving-that-if-the-dual-space-x-of-a-banach-space-x-is-separable-then%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
$begingroup$
That is false. It should read, the unit sphere, not the unit ball.
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
$endgroup$
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
add a comment |
$begingroup$
That is false. It should read, the unit sphere, not the unit ball.
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
$endgroup$
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
add a comment |
$begingroup$
That is false. It should read, the unit sphere, not the unit ball.
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
$endgroup$
That is false. It should read, the unit sphere, not the unit ball.
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
answered Dec 8 '18 at 13:42
Ben WBen W
2,274615
2,274615
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
add a comment |
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Is it right that : $f $ belongs to the unit sphere means $|f | =1 $ and $f $ belongs to the unit ball means $| f | le 1$ ?
$endgroup$
– J.Guo
Dec 8 '18 at 13:48
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
$begingroup$
Quite right : )
$endgroup$
– Ben W
Dec 8 '18 at 13:49
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%2f3031116%2fproving-that-if-the-dual-space-x-of-a-banach-space-x-is-separable-then%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
1
$begingroup$
I think that both comments and the answer there explain that the proof as written there assumes that $f_n$ belong to the unit sphere, i.e., $|f_n|=1$. Using that, you can get $x_n$ such that $f_n(x_n)gefrac12$.
$endgroup$
– Martin Sleziak
Dec 8 '18 at 13:43