Every proper subspace of a normed linear space is either dense or nowhere dense.

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
1
down vote

favorite














Every proper subspace of a normed linear space is either dense or nowhere dense.





Our instructor proved this theorem in our class. I have understood each and every step of the proof. But still I have a confusion. Confusion arose due the following example $:$



Consider the real line $Bbb R$ with the Euclidean norm. Consider the proper subspace $A=[-1,1] cap Bbb Q$ then this is neither dense nor nowhere dense. But if the example is true that would definitely violate the above theorem. What's going wrong here? Inspite of my effort I couldn't find out why is it happening. Please help me in this regard.



Thank you very much.







share|cite|improve this question
















  • 6




    Subspace in this context is a linear subspace. Your set $A$ is not a vector space.
    – Kavi Rama Murthy
    Aug 14 at 7:29










  • Yeah I have understood sir. Thank you very much for your help.
    – Debabrata Chattopadhyay.
    Aug 14 at 7:56














up vote
1
down vote

favorite














Every proper subspace of a normed linear space is either dense or nowhere dense.





Our instructor proved this theorem in our class. I have understood each and every step of the proof. But still I have a confusion. Confusion arose due the following example $:$



Consider the real line $Bbb R$ with the Euclidean norm. Consider the proper subspace $A=[-1,1] cap Bbb Q$ then this is neither dense nor nowhere dense. But if the example is true that would definitely violate the above theorem. What's going wrong here? Inspite of my effort I couldn't find out why is it happening. Please help me in this regard.



Thank you very much.







share|cite|improve this question
















  • 6




    Subspace in this context is a linear subspace. Your set $A$ is not a vector space.
    – Kavi Rama Murthy
    Aug 14 at 7:29










  • Yeah I have understood sir. Thank you very much for your help.
    – Debabrata Chattopadhyay.
    Aug 14 at 7:56












up vote
1
down vote

favorite









up vote
1
down vote

favorite













Every proper subspace of a normed linear space is either dense or nowhere dense.





Our instructor proved this theorem in our class. I have understood each and every step of the proof. But still I have a confusion. Confusion arose due the following example $:$



Consider the real line $Bbb R$ with the Euclidean norm. Consider the proper subspace $A=[-1,1] cap Bbb Q$ then this is neither dense nor nowhere dense. But if the example is true that would definitely violate the above theorem. What's going wrong here? Inspite of my effort I couldn't find out why is it happening. Please help me in this regard.



Thank you very much.







share|cite|improve this question














Every proper subspace of a normed linear space is either dense or nowhere dense.





Our instructor proved this theorem in our class. I have understood each and every step of the proof. But still I have a confusion. Confusion arose due the following example $:$



Consider the real line $Bbb R$ with the Euclidean norm. Consider the proper subspace $A=[-1,1] cap Bbb Q$ then this is neither dense nor nowhere dense. But if the example is true that would definitely violate the above theorem. What's going wrong here? Inspite of my effort I couldn't find out why is it happening. Please help me in this regard.



Thank you very much.









share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Aug 14 at 7:26









Debabrata Chattopadhyay.

15612




15612







  • 6




    Subspace in this context is a linear subspace. Your set $A$ is not a vector space.
    – Kavi Rama Murthy
    Aug 14 at 7:29










  • Yeah I have understood sir. Thank you very much for your help.
    – Debabrata Chattopadhyay.
    Aug 14 at 7:56












  • 6




    Subspace in this context is a linear subspace. Your set $A$ is not a vector space.
    – Kavi Rama Murthy
    Aug 14 at 7:29










  • Yeah I have understood sir. Thank you very much for your help.
    – Debabrata Chattopadhyay.
    Aug 14 at 7:56







6




6




Subspace in this context is a linear subspace. Your set $A$ is not a vector space.
– Kavi Rama Murthy
Aug 14 at 7:29




Subspace in this context is a linear subspace. Your set $A$ is not a vector space.
– Kavi Rama Murthy
Aug 14 at 7:29












Yeah I have understood sir. Thank you very much for your help.
– Debabrata Chattopadhyay.
Aug 14 at 7:56




Yeah I have understood sir. Thank you very much for your help.
– Debabrata Chattopadhyay.
Aug 14 at 7:56















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',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
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%2f2882162%2fevery-proper-subspace-of-a-normed-linear-space-is-either-dense-or-nowhere-dense%23new-answer', 'question_page');

);

Post as a guest



































active

oldest

votes













active

oldest

votes









active

oldest

votes






active

oldest

votes










 

draft saved


draft discarded


























 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2882162%2fevery-proper-subspace-of-a-normed-linear-space-is-either-dense-or-nowhere-dense%23new-answer', 'question_page');

);

Post as a guest













































































這個網誌中的熱門文章

How to combine Bézier curves to a surface?

Carbon dioxide

Why am i infinitely getting the same tweet with the Twitter Search API?