Normed spaces functions?












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Ive just started to learn about normed spaces and usually the definition requires a vector v to be in the vector space V and thus they define the 2-norm for example, to be the sum of the root of the sum of the squares from 1 to n. However ive come across some questions where v is a function with a real domain so how it doesnt make sense if i use the sum definition above for the 2-norm since the domain isnt discrete. Would it be integral instead? Same with the sup-norm.










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    The $L_2$ norm is $|f|_2:=sqrt{int_X|f|^2,dmu},$ where we are using the Lebesgue integral. There are other norms defined analogously.
    $endgroup$
    – Adrian Keister
    Dec 10 '18 at 20:13












  • $begingroup$
    The definition you've seen of the $2$-norm was never supposed to make sense for arbitrary vector spaces; it is specifically about $mathbb{R}^n$ and $mathbb{C}^n$, so that you can meaningfully refer to coordinates.
    $endgroup$
    – Qiaochu Yuan
    Dec 11 '18 at 0:57
















0












$begingroup$


Ive just started to learn about normed spaces and usually the definition requires a vector v to be in the vector space V and thus they define the 2-norm for example, to be the sum of the root of the sum of the squares from 1 to n. However ive come across some questions where v is a function with a real domain so how it doesnt make sense if i use the sum definition above for the 2-norm since the domain isnt discrete. Would it be integral instead? Same with the sup-norm.










share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    The $L_2$ norm is $|f|_2:=sqrt{int_X|f|^2,dmu},$ where we are using the Lebesgue integral. There are other norms defined analogously.
    $endgroup$
    – Adrian Keister
    Dec 10 '18 at 20:13












  • $begingroup$
    The definition you've seen of the $2$-norm was never supposed to make sense for arbitrary vector spaces; it is specifically about $mathbb{R}^n$ and $mathbb{C}^n$, so that you can meaningfully refer to coordinates.
    $endgroup$
    – Qiaochu Yuan
    Dec 11 '18 at 0:57














0












0








0





$begingroup$


Ive just started to learn about normed spaces and usually the definition requires a vector v to be in the vector space V and thus they define the 2-norm for example, to be the sum of the root of the sum of the squares from 1 to n. However ive come across some questions where v is a function with a real domain so how it doesnt make sense if i use the sum definition above for the 2-norm since the domain isnt discrete. Would it be integral instead? Same with the sup-norm.










share|cite|improve this question









$endgroup$




Ive just started to learn about normed spaces and usually the definition requires a vector v to be in the vector space V and thus they define the 2-norm for example, to be the sum of the root of the sum of the squares from 1 to n. However ive come across some questions where v is a function with a real domain so how it doesnt make sense if i use the sum definition above for the 2-norm since the domain isnt discrete. Would it be integral instead? Same with the sup-norm.







functional-analysis






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share|cite|improve this question











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asked Dec 10 '18 at 20:11









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  • 1




    $begingroup$
    The $L_2$ norm is $|f|_2:=sqrt{int_X|f|^2,dmu},$ where we are using the Lebesgue integral. There are other norms defined analogously.
    $endgroup$
    – Adrian Keister
    Dec 10 '18 at 20:13












  • $begingroup$
    The definition you've seen of the $2$-norm was never supposed to make sense for arbitrary vector spaces; it is specifically about $mathbb{R}^n$ and $mathbb{C}^n$, so that you can meaningfully refer to coordinates.
    $endgroup$
    – Qiaochu Yuan
    Dec 11 '18 at 0:57














  • 1




    $begingroup$
    The $L_2$ norm is $|f|_2:=sqrt{int_X|f|^2,dmu},$ where we are using the Lebesgue integral. There are other norms defined analogously.
    $endgroup$
    – Adrian Keister
    Dec 10 '18 at 20:13












  • $begingroup$
    The definition you've seen of the $2$-norm was never supposed to make sense for arbitrary vector spaces; it is specifically about $mathbb{R}^n$ and $mathbb{C}^n$, so that you can meaningfully refer to coordinates.
    $endgroup$
    – Qiaochu Yuan
    Dec 11 '18 at 0:57








1




1




$begingroup$
The $L_2$ norm is $|f|_2:=sqrt{int_X|f|^2,dmu},$ where we are using the Lebesgue integral. There are other norms defined analogously.
$endgroup$
– Adrian Keister
Dec 10 '18 at 20:13






$begingroup$
The $L_2$ norm is $|f|_2:=sqrt{int_X|f|^2,dmu},$ where we are using the Lebesgue integral. There are other norms defined analogously.
$endgroup$
– Adrian Keister
Dec 10 '18 at 20:13














$begingroup$
The definition you've seen of the $2$-norm was never supposed to make sense for arbitrary vector spaces; it is specifically about $mathbb{R}^n$ and $mathbb{C}^n$, so that you can meaningfully refer to coordinates.
$endgroup$
– Qiaochu Yuan
Dec 11 '18 at 0:57




$begingroup$
The definition you've seen of the $2$-norm was never supposed to make sense for arbitrary vector spaces; it is specifically about $mathbb{R}^n$ and $mathbb{C}^n$, so that you can meaningfully refer to coordinates.
$endgroup$
– Qiaochu Yuan
Dec 11 '18 at 0:57










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