Ttykuu

A newer friend of my brother's gave him a load of baseball cards that are supposedly extremely valuable. Is this a scam?

How is it possible for user to changed after storage was encrypted? (on OS X, Android)

whey we use polarized capacitor?

How can the DM most effectively choose 1 out of an odd number of players to be targeted by an attack or effect?

Find original functions from a composite function

Pronouncing Dictionary.com's W.O.D "vade mecum" in English

Patience, young "Padovan"

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

Infinite past with a beginning?

What are these boxed doors outside store fronts in New York?

Banach space and Hilbert space topology

Why don't electron-positron collisions release infinite energy?

Why don't electromagnetic waves interact with each other?

Why are only specific transaction types accepted into the mempool?

Validation accuracy vs Testing accuracy

What typically incentivizes a professor to change jobs to a lower ranking university?

How to type dʒ symbol (IPA) on Mac?

If Manufacturer spice model and Datasheet give different values which should I use?

GPS Rollover on Android Smartphones

Symplectic equivalent of commuting matrices

How do we improve the relationship with a client software team that performs poorly and is becoming less collaborative?

Do Phineas and Ferb ever actually get busted in real time?

How did the USSR manage to innovate in an environment characterized by government censorship and high bureaucracy?

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

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

edited 1 hour ago

Henno Brandsma

115k349125

asked 1 hour ago

user156213user156213

60338

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

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

edited 1 hour ago

Henno Brandsma

115k349125

asked 1 hour ago

user156213user156213

60338

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

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

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

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

  
  
  
  

add a comment | 

$begingroup$

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

edited 1 hour ago

Henno Brandsma

115k349125

asked 1 hour ago

user156213user156213

60338

$endgroup$

share|cite|improve this question

edited 1 hour ago

Henno Brandsma

115k349125

asked 1 hour ago

user156213user156213

60338

share|cite|improve this question

edited 1 hour ago

Henno Brandsma

115k349125

asked 1 hour ago

user156213user156213

60338

edited 1 hour ago

Henno Brandsma

115k349125

edited 1 hour ago

Henno Brandsma

115k349125

asked 1 hour ago

user156213user156213

60338

asked 1 hour ago

user156213user156213

60338

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

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

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
);



);

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');

var $window = $(window),
onScroll = function(e)
var $elem = $('.new-login-left'),
docViewTop = $window.scrollTop(),
docViewBottom = docViewTop + $window.height(),
elemTop = $elem.offset().top,
elemBottom = elemTop + $elem.height();
if ((docViewTop elemBottom))
StackExchange.using('gps', function() StackExchange.gps.track('embedded_signup_form.view', location: 'question_page' ); );
$window.unbind('scroll', onScroll);

;
$window.on('scroll', onScroll);

);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name

Email

Required, but never shown

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

Name

Email

Required, but never shown

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

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

add a comment | 

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

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

add a comment | 

$begingroup$

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

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125

$endgroup$

share|cite|improve this answer

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125

answered 1 hour ago

Henno BrandsmaHenno Brandsma

115k349125