Notations in Functional Analysis: $L^p$, $L_p$, $mathscr{L}^p$, $mathscr{L}_p$, $mathcal{L}^p$, and...
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If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$-norm (i.e, $p^text{th}$-power-integrable functions), while other use the notation $L^p$ for the same purpose. If you are a functional analyst that distinguishes between $L^p$ and $L_p$, then could you please let me know what $L_p$ means (presumably, your definition of $L^p$ coincides with Wikipedia's definition)?
Now, I found another similar notation $mathscr{L}_p$. What is $mathscr{L}_p$? There seem to be $mathscr{L}^p$, $mathcal{L}^p$, and $mathcal{L}_p$ too. However, I would expect that this is just due to people's using different fonts, i.e., $L^p=mathscr{L}^p=mathcal{L}^p$ and $L_p=mathscr{L}_p=mathcal{L}_p$.
functional-analysis soft-question notation definition lp-spaces
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
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add a comment |
$begingroup$
If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$-norm (i.e, $p^text{th}$-power-integrable functions), while other use the notation $L^p$ for the same purpose. If you are a functional analyst that distinguishes between $L^p$ and $L_p$, then could you please let me know what $L_p$ means (presumably, your definition of $L^p$ coincides with Wikipedia's definition)?
Now, I found another similar notation $mathscr{L}_p$. What is $mathscr{L}_p$? There seem to be $mathscr{L}^p$, $mathcal{L}^p$, and $mathcal{L}_p$ too. However, I would expect that this is just due to people's using different fonts, i.e., $L^p=mathscr{L}^p=mathcal{L}^p$ and $L_p=mathscr{L}_p=mathcal{L}_p$.
functional-analysis soft-question notation definition lp-spaces
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
$endgroup$
3
$begingroup$
I have seen $L_p$ being used as the space containing actual functions with finite $Vert cdotVert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is.
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– Jan Bohr
Dec 11 '18 at 21:20
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Perhaps you're thinking of the distinction between $L_p$ spaces and $mathscr L_p$ spaces.
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– Robert Israel
Dec 11 '18 at 21:27
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@RobertIsrael What are $L_p$ and $mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.)
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– Batominovski
Dec 11 '18 at 21:28
4
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I learned it like this (from Dirk Werner personally :) )... $mathscr L^p$ denotes the space of functions with finite $||,cdot,||_p$-semi norm and $L^p$ is the space of equivalence classes $f sim g$ if $f=g$ almost everywhere. The classes are such that $||,cdot,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols.
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– N.Beck
Dec 11 '18 at 22:07
add a comment |
$begingroup$
If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$-norm (i.e, $p^text{th}$-power-integrable functions), while other use the notation $L^p$ for the same purpose. If you are a functional analyst that distinguishes between $L^p$ and $L_p$, then could you please let me know what $L_p$ means (presumably, your definition of $L^p$ coincides with Wikipedia's definition)?
Now, I found another similar notation $mathscr{L}_p$. What is $mathscr{L}_p$? There seem to be $mathscr{L}^p$, $mathcal{L}^p$, and $mathcal{L}_p$ too. However, I would expect that this is just due to people's using different fonts, i.e., $L^p=mathscr{L}^p=mathcal{L}^p$ and $L_p=mathscr{L}_p=mathcal{L}_p$.
functional-analysis soft-question notation definition lp-spaces
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
$endgroup$
If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$-norm (i.e, $p^text{th}$-power-integrable functions), while other use the notation $L^p$ for the same purpose. If you are a functional analyst that distinguishes between $L^p$ and $L_p$, then could you please let me know what $L_p$ means (presumably, your definition of $L^p$ coincides with Wikipedia's definition)?
Now, I found another similar notation $mathscr{L}_p$. What is $mathscr{L}_p$? There seem to be $mathscr{L}^p$, $mathcal{L}^p$, and $mathcal{L}_p$ too. However, I would expect that this is just due to people's using different fonts, i.e., $L^p=mathscr{L}^p=mathcal{L}^p$ and $L_p=mathscr{L}_p=mathcal{L}_p$.
functional-analysis soft-question notation definition lp-spaces
functional-analysis soft-question notation definition lp-spaces
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
edited Dec 12 '18 at 13:15
Batominovski
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
asked Dec 11 '18 at 21:03
BatominovskiBatominovski
33.1k33293
33.1k33293
3
$begingroup$
I have seen $L_p$ being used as the space containing actual functions with finite $Vert cdotVert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is.
$endgroup$
– Jan Bohr
Dec 11 '18 at 21:20
$begingroup$
Perhaps you're thinking of the distinction between $L_p$ spaces and $mathscr L_p$ spaces.
$endgroup$
– Robert Israel
Dec 11 '18 at 21:27
$begingroup$
@RobertIsrael What are $L_p$ and $mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.)
$endgroup$
– Batominovski
Dec 11 '18 at 21:28
4
$begingroup$
I learned it like this (from Dirk Werner personally :) )... $mathscr L^p$ denotes the space of functions with finite $||,cdot,||_p$-semi norm and $L^p$ is the space of equivalence classes $f sim g$ if $f=g$ almost everywhere. The classes are such that $||,cdot,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols.
$endgroup$
– N.Beck
Dec 11 '18 at 22:07
add a comment |
3
$begingroup$
I have seen $L_p$ being used as the space containing actual functions with finite $Vert cdotVert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is.
$endgroup$
– Jan Bohr
Dec 11 '18 at 21:20
$begingroup$
Perhaps you're thinking of the distinction between $L_p$ spaces and $mathscr L_p$ spaces.
$endgroup$
– Robert Israel
Dec 11 '18 at 21:27
$begingroup$
@RobertIsrael What are $L_p$ and $mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.)
$endgroup$
– Batominovski
Dec 11 '18 at 21:28
4
$begingroup$
I learned it like this (from Dirk Werner personally :) )... $mathscr L^p$ denotes the space of functions with finite $||,cdot,||_p$-semi norm and $L^p$ is the space of equivalence classes $f sim g$ if $f=g$ almost everywhere. The classes are such that $||,cdot,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols.
$endgroup$
– N.Beck
Dec 11 '18 at 22:07
3
3
$begingroup$
I have seen $L_p$ being used as the space containing actual functions with finite $Vert cdotVert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is.
$endgroup$
– Jan Bohr
Dec 11 '18 at 21:20
$begingroup$
I have seen $L_p$ being used as the space containing actual functions with finite $Vert cdotVert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is.
$endgroup$
– Jan Bohr
Dec 11 '18 at 21:20
$begingroup$
Perhaps you're thinking of the distinction between $L_p$ spaces and $mathscr L_p$ spaces.
$endgroup$
– Robert Israel
Dec 11 '18 at 21:27
$begingroup$
Perhaps you're thinking of the distinction between $L_p$ spaces and $mathscr L_p$ spaces.
$endgroup$
– Robert Israel
Dec 11 '18 at 21:27
$begingroup$
@RobertIsrael What are $L_p$ and $mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.)
$endgroup$
– Batominovski
Dec 11 '18 at 21:28
$begingroup$
@RobertIsrael What are $L_p$ and $mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.)
$endgroup$
– Batominovski
Dec 11 '18 at 21:28
4
4
$begingroup$
I learned it like this (from Dirk Werner personally :) )... $mathscr L^p$ denotes the space of functions with finite $||,cdot,||_p$-semi norm and $L^p$ is the space of equivalence classes $f sim g$ if $f=g$ almost everywhere. The classes are such that $||,cdot,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols.
$endgroup$
– N.Beck
Dec 11 '18 at 22:07
$begingroup$
I learned it like this (from Dirk Werner personally :) )... $mathscr L^p$ denotes the space of functions with finite $||,cdot,||_p$-semi norm and $L^p$ is the space of equivalence classes $f sim g$ if $f=g$ almost everywhere. The classes are such that $||,cdot,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols.
$endgroup$
– N.Beck
Dec 11 '18 at 22:07
add a comment |
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$begingroup$
I have seen $L_p$ being used as the space containing actual functions with finite $Vert cdotVert_p-$norm, as opposed to $L^p$ being the space of a.e.-equivalence classes of such functions. But I don't know how common this is.
$endgroup$
– Jan Bohr
Dec 11 '18 at 21:20
$begingroup$
Perhaps you're thinking of the distinction between $L_p$ spaces and $mathscr L_p$ spaces.
$endgroup$
– Robert Israel
Dec 11 '18 at 21:27
$begingroup$
@RobertIsrael What are $L_p$ and $mathscr{L}_p$ in you case? (It is difficult to do a Google search on this, when Google recognizes any "Lp space" as "$L^p$ space" and nothing else.)
$endgroup$
– Batominovski
Dec 11 '18 at 21:28
4
$begingroup$
I learned it like this (from Dirk Werner personally :) )... $mathscr L^p$ denotes the space of functions with finite $||,cdot,||_p$-semi norm and $L^p$ is the space of equivalence classes $f sim g$ if $f=g$ almost everywhere. The classes are such that $||,cdot,||_p$ becomes a norm. I don't think it makes any sence to have more than two different symbols.
$endgroup$
– N.Beck
Dec 11 '18 at 22:07