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







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  • 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


















1












$begingroup$


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







share|cite|improve this question











$endgroup$








  • 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.
    $endgroup$
    – 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
















1












1








1





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 12 '18 at 13:15







Batominovski

















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
















  • 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












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