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.
functional-analysis
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
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add a comment |
$begingroup$
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.
functional-analysis
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
$endgroup$
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.
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– Adrian Keister
Dec 10 '18 at 20:13
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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.
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– Qiaochu Yuan
Dec 11 '18 at 0:57
add a comment |
$begingroup$
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.
functional-analysis
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
$endgroup$
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
functional-analysis
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
asked Dec 10 '18 at 20:11
NoteBookNoteBook
1167
1167
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
add a comment |
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
add a comment |
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$begingroup$
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