Banach space and Hilbert space topologyIs any Banach space a dual space?A Banach space that is not a Hilbert spaceIs every Hilbert space a Banach algebra?Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.Is every Banach space densely embedded in a Hilbert space?Existence of a $mathbb C$-Banach space isometric to a Hilbert Space but whose norm is not induced by an inner product?An example of a Banach space isomorphic but not isometric to a dual Banach spaceThe Hahn-Banach Theorem for Hilbert SpaceBanach spaces and Hilbert spaceBasis of infinite dimensional Banach space and separable hilbert space

How do you conduct xenoanthropology after first contact?

How does one intimidate enemies without having the capacity for violence?

Why has Russell's definition of numbers using equivalence classes been finally abandoned? ( If it has actually been abandoned).

Extreme, but not acceptable situation and I can't start the work tomorrow morning

Is there a minimum number of transactions in a block?

Why doesn't Newton's third law mean a person bounces back to where they started when they hit the ground?

Download, install and reboot computer at night if needed

Shell script can be run only with sh command

Why did the Germans forbid the possession of pet pigeons in Rostov-on-Don in 1941?

DOS, create pipe for stdin/stdout of command.com(or 4dos.com) in C or Batch?

A Journey Through Space and Time

Accidentally leaked the solution to an assignment, what to do now? (I'm the prof)

"which" command doesn't work / path of Safari?

declaring a variable twice in IIFE

Can Medicine checks be used, with decent rolls, to completely mitigate the risk of death from ongoing damage?

How can bays and straits be determined in a procedurally generated map?

Should I join an office cleaning event for free?

XeLaTeX and pdfLaTeX ignore hyphenation

How is this relation reflexive?

How old can references or sources in a thesis be?

Is it possible to do 50 km distance without any previous training?

Modification to Chariots for Heavy Cavalry Analogue for 4-armed race

Why is this code 6.5x slower with optimizations enabled?

What defenses are there against being summoned by the Gate spell?



Banach space and Hilbert space topology


Is any Banach space a dual space?A Banach space that is not a Hilbert spaceIs every Hilbert space a Banach algebra?Which Hilbert space is isometrically isomorphism with $B(E)$ for some Banach space $E$.Is every Banach space densely embedded in a Hilbert space?Existence of a $mathbb C$-Banach space isometric to a Hilbert Space but whose norm is not induced by an inner product?An example of a Banach space isomorphic but not isometric to a dual Banach spaceThe Hahn-Banach Theorem for Hilbert SpaceBanach spaces and Hilbert spaceBasis of infinite dimensional Banach space and separable hilbert space













1












$begingroup$


Let $B$ be a Banach space. It is not necessarily true that
there exists a Hilbert space $H$ linearly isometric to $B$.



However, is it true that there exists a Hilbert space $H$
homeomorphic to $B$?






general-topology functional-analysis hilbert-spaces banach-spaces







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    If $B$ is separable, then yes. All separable Banach Spaces are homeomorphic. So homeomorphic to $ell^2$
    $endgroup$
    – user124910
    6 hours ago






  • 1




    $begingroup$
    @user124910 We can extend this to non-separable as well. See my answer.
    $endgroup$
    – Henno Brandsma
    6 hours ago















1












$begingroup$


Let $B$ be a Banach space. It is not necessarily true that
there exists a Hilbert space $H$ linearly isometric to $B$.



However, is it true that there exists a Hilbert space $H$
homeomorphic to $B$?






general-topology functional-analysis hilbert-spaces banach-spaces







share|cite|improve this question











$endgroup$







  • 1




    $begingroup$
    If $B$ is separable, then yes. All separable Banach Spaces are homeomorphic. So homeomorphic to $ell^2$
    $endgroup$
    – user124910
    6 hours ago






  • 1




    $begingroup$
    @user124910 We can extend this to non-separable as well. See my answer.
    $endgroup$
    – Henno Brandsma
    6 hours ago













1












1








1





$begingroup$


Let $B$ be a Banach space. It is not necessarily true that
there exists a Hilbert space $H$ linearly isometric to $B$.



However, is it true that there exists a Hilbert space $H$
homeomorphic to $B$?






general-topology functional-analysis hilbert-spaces banach-spaces







share|cite|improve this question











$endgroup$




Let $B$ be a Banach space. It is not necessarily true that
there exists a Hilbert space $H$ linearly isometric to $B$.



However, is it true that there exists a Hilbert space $H$
homeomorphic to $B$?






general-topology functional-analysis hilbert-spaces banach-spaces




general-topology functional-analysis hilbert-spaces banach-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 6 hours ago









Henno Brandsma

115k349125




115k349125










asked 6 hours ago









user156213user156213

60338




60338







  • 1




    $begingroup$
    If $B$ is separable, then yes. All separable Banach Spaces are homeomorphic. So homeomorphic to $ell^2$
    $endgroup$
    – user124910
    6 hours ago






  • 1




    $begingroup$
    @user124910 We can extend this to non-separable as well. See my answer.
    $endgroup$
    – Henno Brandsma
    6 hours ago












  • 1




    $begingroup$
    If $B$ is separable, then yes. All separable Banach Spaces are homeomorphic. So homeomorphic to $ell^2$
    $endgroup$
    – user124910
    6 hours ago






  • 1




    $begingroup$
    @user124910 We can extend this to non-separable as well. See my answer.
    $endgroup$
    – Henno Brandsma
    6 hours ago







1




1




$begingroup$
If $B$ is separable, then yes. All separable Banach Spaces are homeomorphic. So homeomorphic to $ell^2$
$endgroup$
– user124910
6 hours ago




$begingroup$
If $B$ is separable, then yes. All separable Banach Spaces are homeomorphic. So homeomorphic to $ell^2$
$endgroup$
– user124910
6 hours ago




1




1




$begingroup$
@user124910 We can extend this to non-separable as well. See my answer.
$endgroup$
– Henno Brandsma
6 hours ago




$begingroup$
@user124910 We can extend this to non-separable as well. See my answer.
$endgroup$
– Henno Brandsma
6 hours ago










1 Answer
1






active

oldest

votes


















5












$begingroup$

Yes, but this is quite a deep result. Two infinite-dimensional Banach spaces $X$ and $Y$ are homeomorphic iff $d(X)=d(Y)$, where the density $d(X)$ is the minimal size of a dense subset of $X$.



So any separable infinite-dimensional Banach space is homeomorphic to the Hilbert space $ell^2$ (and even to $mathbbR^omega$, because the result extends to locally convex completely metrisable TVS's as well). And for higher densities we have Hilbert spaces $ell_2(kappa)$ as models. Finite dimensional we only have the $mathbbR^n$ up to homeomorphism, which are already Hilbert spaces.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Do you know of a reference with the proof of this?
    $endgroup$
    – user156213
    3 hours ago











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%2f3178808%2fbanach-space-and-hilbert-space-topology%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









5












$begingroup$

Yes, but this is quite a deep result. Two infinite-dimensional Banach spaces $X$ and $Y$ are homeomorphic iff $d(X)=d(Y)$, where the density $d(X)$ is the minimal size of a dense subset of $X$.



So any separable infinite-dimensional Banach space is homeomorphic to the Hilbert space $ell^2$ (and even to $mathbbR^omega$, because the result extends to locally convex completely metrisable TVS's as well). And for higher densities we have Hilbert spaces $ell_2(kappa)$ as models. Finite dimensional we only have the $mathbbR^n$ up to homeomorphism, which are already Hilbert spaces.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Do you know of a reference with the proof of this?
    $endgroup$
    – user156213
    3 hours ago















5












$begingroup$

Yes, but this is quite a deep result. Two infinite-dimensional Banach spaces $X$ and $Y$ are homeomorphic iff $d(X)=d(Y)$, where the density $d(X)$ is the minimal size of a dense subset of $X$.



So any separable infinite-dimensional Banach space is homeomorphic to the Hilbert space $ell^2$ (and even to $mathbbR^omega$, because the result extends to locally convex completely metrisable TVS's as well). And for higher densities we have Hilbert spaces $ell_2(kappa)$ as models. Finite dimensional we only have the $mathbbR^n$ up to homeomorphism, which are already Hilbert spaces.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Do you know of a reference with the proof of this?
    $endgroup$
    – user156213
    3 hours ago













5












5








5





$begingroup$

Yes, but this is quite a deep result. Two infinite-dimensional Banach spaces $X$ and $Y$ are homeomorphic iff $d(X)=d(Y)$, where the density $d(X)$ is the minimal size of a dense subset of $X$.



So any separable infinite-dimensional Banach space is homeomorphic to the Hilbert space $ell^2$ (and even to $mathbbR^omega$, because the result extends to locally convex completely metrisable TVS's as well). And for higher densities we have Hilbert spaces $ell_2(kappa)$ as models. Finite dimensional we only have the $mathbbR^n$ up to homeomorphism, which are already Hilbert spaces.






share|cite|improve this answer









$endgroup$



Yes, but this is quite a deep result. Two infinite-dimensional Banach spaces $X$ and $Y$ are homeomorphic iff $d(X)=d(Y)$, where the density $d(X)$ is the minimal size of a dense subset of $X$.



So any separable infinite-dimensional Banach space is homeomorphic to the Hilbert space $ell^2$ (and even to $mathbbR^omega$, because the result extends to locally convex completely metrisable TVS's as well). And for higher densities we have Hilbert spaces $ell_2(kappa)$ as models. Finite dimensional we only have the $mathbbR^n$ up to homeomorphism, which are already Hilbert spaces.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 6 hours ago









Henno BrandsmaHenno Brandsma

115k349125




115k349125











  • $begingroup$
    Do you know of a reference with the proof of this?
    $endgroup$
    – user156213
    3 hours ago
















  • $begingroup$
    Do you know of a reference with the proof of this?
    $endgroup$
    – user156213
    3 hours ago















$begingroup$
Do you know of a reference with the proof of this?
$endgroup$
– user156213
3 hours ago




$begingroup$
Do you know of a reference with the proof of this?
$endgroup$
– user156213
3 hours ago

















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%2f3178808%2fbanach-space-and-hilbert-space-topology%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

منجزی محتویات تیره‌های طایفه منجزی[ویرایش] مشاهیر طایفه منجزی[ویرایش] محل سکونت[ویرایش] پانویس[ویرایش] منابع[ویرایش] منوی ناوبری«نمودار اجتماعی طوایف بختیاری»«Bakhtyārī»«BAḴTĪĀRĪ TRIBE»«اسامی طوایف و شعب ایل بختیاری»ووگسترش آن

Image
مقاله‌های خرد ایل بختیاریایل بختیاریبختیاریبهداروند بابهفت لنگطایفه‌های بختیاری ایل بختیاریهفت لنگبهداروندبختیارونداصفهانچهارمحال بختیاریخوزستانتیرهحاتم ...
Read more

بیوانفورماتیک محتویات تاریخچه[ویرایش] اهداف[ویرایش] کاربردهای بیوانفورماتیک[ویرایش] زمینه‌های مهم بیوانفورماتیک[ویرایش] موضوعات سیستم نرم‌افزاری بیوانفورماتیک[ویرایش] مراکز و ابزار[ویرایش] جستارهای وابسته[ویرایش] پیوند به بیرون[ویرایش] منابع[ویرایش] منوی ناوبریووانجمن بیوانفورماتیک ایرانمرکز بیوانفورماتیک دانشگاه تهرانمرکز ملی تحقیقات بیوانفورماتیکانستیتو بیو-آی‌تیبانک داده‌های دی ان ایمرکز ملی اطلاعات بیوتکنولوژیانستیتوی بیوانفورماتیک اروپاپورتال بیوانفورماتیک ایران«فرهنگ واژه‌های مصوب فرهنگستان ـ دفتر هشتم، بخش لاتین»وووو4611085-900870420ووو

Image
منطق ریاضینظریه مجموعه‌هانظریه اعدادنظریه گرافنظریه نوع‌هانظریه رده‌هاآنالیز عددینظریه اطلاعاتترکیبیاتجبر بولیتحلیل الگوریتم‌هاطراحی الگوریتمبهینه‌سازی ت...
Read more