“nice functions”












4












$begingroup$


I see the statement of "nice functions" in textbooks and the authors usually don't need to give the definition of "nice functions". For example in a book which I read now the authors write



"Morrey spaces is not separable. A version of Morrey space where it is possible to approximate by "nice functions" is vanishing Morrey space."



and don't give the definition of "nice functions" anywhere in the book.



I wonder in here what is the meaning of "nice functions" ?



and



Is there a fixed definition of "nice functions" ?






analysis functions







share|cite|improve this question









$endgroup$












  • $begingroup$
    It means a function that has sufficiently many derivatives, and which converges fast enough at infinity, that we don't have to worry about rigour when considering any expression involving derivatives, integrals, or integration by parts. Usually you can replace "nice" by "Schwarz class." But it has no real definition, and the term is intentionally vague.
    $endgroup$
    – Stephen Montgomery-Smith
    Apr 22 '14 at 12:11








  • 7




    $begingroup$
    I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
    $endgroup$
    – apnorton
    Apr 22 '14 at 14:01










  • $begingroup$
    @anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
    $endgroup$
    – Mario Carneiro
    Apr 22 '14 at 16:52










  • $begingroup$
    @MarioCarneiro Done.
    $endgroup$
    – apnorton
    Apr 22 '14 at 18:16
















4












$begingroup$


I see the statement of "nice functions" in textbooks and the authors usually don't need to give the definition of "nice functions". For example in a book which I read now the authors write



"Morrey spaces is not separable. A version of Morrey space where it is possible to approximate by "nice functions" is vanishing Morrey space."



and don't give the definition of "nice functions" anywhere in the book.



I wonder in here what is the meaning of "nice functions" ?



and



Is there a fixed definition of "nice functions" ?






analysis functions







share|cite|improve this question









$endgroup$












  • $begingroup$
    It means a function that has sufficiently many derivatives, and which converges fast enough at infinity, that we don't have to worry about rigour when considering any expression involving derivatives, integrals, or integration by parts. Usually you can replace "nice" by "Schwarz class." But it has no real definition, and the term is intentionally vague.
    $endgroup$
    – Stephen Montgomery-Smith
    Apr 22 '14 at 12:11








  • 7




    $begingroup$
    I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
    $endgroup$
    – apnorton
    Apr 22 '14 at 14:01










  • $begingroup$
    @anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
    $endgroup$
    – Mario Carneiro
    Apr 22 '14 at 16:52










  • $begingroup$
    @MarioCarneiro Done.
    $endgroup$
    – apnorton
    Apr 22 '14 at 18:16














4












4








4


2



$begingroup$


I see the statement of "nice functions" in textbooks and the authors usually don't need to give the definition of "nice functions". For example in a book which I read now the authors write



"Morrey spaces is not separable. A version of Morrey space where it is possible to approximate by "nice functions" is vanishing Morrey space."



and don't give the definition of "nice functions" anywhere in the book.



I wonder in here what is the meaning of "nice functions" ?



and



Is there a fixed definition of "nice functions" ?






analysis functions







share|cite|improve this question









$endgroup$




I see the statement of "nice functions" in textbooks and the authors usually don't need to give the definition of "nice functions". For example in a book which I read now the authors write



"Morrey spaces is not separable. A version of Morrey space where it is possible to approximate by "nice functions" is vanishing Morrey space."



and don't give the definition of "nice functions" anywhere in the book.



I wonder in here what is the meaning of "nice functions" ?



and



Is there a fixed definition of "nice functions" ?






analysis functions




analysis functions






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Apr 22 '14 at 12:02









bjk1806bjk1806

165112




165112












  • $begingroup$
    It means a function that has sufficiently many derivatives, and which converges fast enough at infinity, that we don't have to worry about rigour when considering any expression involving derivatives, integrals, or integration by parts. Usually you can replace "nice" by "Schwarz class." But it has no real definition, and the term is intentionally vague.
    $endgroup$
    – Stephen Montgomery-Smith
    Apr 22 '14 at 12:11








  • 7




    $begingroup$
    I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
    $endgroup$
    – apnorton
    Apr 22 '14 at 14:01










  • $begingroup$
    @anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
    $endgroup$
    – Mario Carneiro
    Apr 22 '14 at 16:52










  • $begingroup$
    @MarioCarneiro Done.
    $endgroup$
    – apnorton
    Apr 22 '14 at 18:16


















  • $begingroup$
    It means a function that has sufficiently many derivatives, and which converges fast enough at infinity, that we don't have to worry about rigour when considering any expression involving derivatives, integrals, or integration by parts. Usually you can replace "nice" by "Schwarz class." But it has no real definition, and the term is intentionally vague.
    $endgroup$
    – Stephen Montgomery-Smith
    Apr 22 '14 at 12:11








  • 7




    $begingroup$
    I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
    $endgroup$
    – apnorton
    Apr 22 '14 at 14:01










  • $begingroup$
    @anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
    $endgroup$
    – Mario Carneiro
    Apr 22 '14 at 16:52










  • $begingroup$
    @MarioCarneiro Done.
    $endgroup$
    – apnorton
    Apr 22 '14 at 18:16
















$begingroup$
It means a function that has sufficiently many derivatives, and which converges fast enough at infinity, that we don't have to worry about rigour when considering any expression involving derivatives, integrals, or integration by parts. Usually you can replace "nice" by "Schwarz class." But it has no real definition, and the term is intentionally vague.
$endgroup$
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11






$begingroup$
It means a function that has sufficiently many derivatives, and which converges fast enough at infinity, that we don't have to worry about rigour when considering any expression involving derivatives, integrals, or integration by parts. Usually you can replace "nice" by "Schwarz class." But it has no real definition, and the term is intentionally vague.
$endgroup$
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11






7




7




$begingroup$
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
$endgroup$
– apnorton
Apr 22 '14 at 14:01




$begingroup$
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
$endgroup$
– apnorton
Apr 22 '14 at 14:01












$begingroup$
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
$endgroup$
– Mario Carneiro
Apr 22 '14 at 16:52




$begingroup$
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
$endgroup$
– Mario Carneiro
Apr 22 '14 at 16:52












$begingroup$
@MarioCarneiro Done.
$endgroup$
– apnorton
Apr 22 '14 at 18:16




$begingroup$
@MarioCarneiro Done.
$endgroup$
– apnorton
Apr 22 '14 at 18:16










3 Answers
3






active

oldest

votes


















11












$begingroup$

The terms "nice" and "good" are used in an ad-hoc way throughout mathematics, and so giving them a fixed definition is counterproductive. The idea is to build intuition: we can expect our theory to work when we only consider objects that are not too strange, or we can guarantee the existence of objects satisfying certain properties that are easy to work with. Concrete definitions do not need to be introduced unless a technical discussion is forthcoming, and can inhibit readability otherwise.






share|cite|improve this answer









$endgroup$





















    7












    $begingroup$

    It might mean continuous, it might mean differentiable, it might mean smooth etc. The only common theme is that the author didn't bother explaining it in more detail.






    share|cite|improve this answer









    $endgroup$





















      6












      $begingroup$

      Comment converted to answer:



      I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."






      share|cite|improve this answer









      $endgroup$













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        3 Answers
        3






        active

        oldest

        votes








        3 Answers
        3






        active

        oldest

        votes









        active

        oldest

        votes






        active

        oldest

        votes









        11












        $begingroup$

        The terms "nice" and "good" are used in an ad-hoc way throughout mathematics, and so giving them a fixed definition is counterproductive. The idea is to build intuition: we can expect our theory to work when we only consider objects that are not too strange, or we can guarantee the existence of objects satisfying certain properties that are easy to work with. Concrete definitions do not need to be introduced unless a technical discussion is forthcoming, and can inhibit readability otherwise.






        share|cite|improve this answer









        $endgroup$


















          11












          $begingroup$

          The terms "nice" and "good" are used in an ad-hoc way throughout mathematics, and so giving them a fixed definition is counterproductive. The idea is to build intuition: we can expect our theory to work when we only consider objects that are not too strange, or we can guarantee the existence of objects satisfying certain properties that are easy to work with. Concrete definitions do not need to be introduced unless a technical discussion is forthcoming, and can inhibit readability otherwise.






          share|cite|improve this answer









          $endgroup$
















            11












            11








            11





            $begingroup$

            The terms "nice" and "good" are used in an ad-hoc way throughout mathematics, and so giving them a fixed definition is counterproductive. The idea is to build intuition: we can expect our theory to work when we only consider objects that are not too strange, or we can guarantee the existence of objects satisfying certain properties that are easy to work with. Concrete definitions do not need to be introduced unless a technical discussion is forthcoming, and can inhibit readability otherwise.






            share|cite|improve this answer









            $endgroup$



            The terms "nice" and "good" are used in an ad-hoc way throughout mathematics, and so giving them a fixed definition is counterproductive. The idea is to build intuition: we can expect our theory to work when we only consider objects that are not too strange, or we can guarantee the existence of objects satisfying certain properties that are easy to work with. Concrete definitions do not need to be introduced unless a technical discussion is forthcoming, and can inhibit readability otherwise.







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Apr 22 '14 at 12:19









            Elchanan SolomonElchanan Solomon

            21.8k54277




            21.8k54277























                7












                $begingroup$

                It might mean continuous, it might mean differentiable, it might mean smooth etc. The only common theme is that the author didn't bother explaining it in more detail.






                share|cite|improve this answer









                $endgroup$


















                  7












                  $begingroup$

                  It might mean continuous, it might mean differentiable, it might mean smooth etc. The only common theme is that the author didn't bother explaining it in more detail.






                  share|cite|improve this answer









                  $endgroup$
















                    7












                    7








                    7





                    $begingroup$

                    It might mean continuous, it might mean differentiable, it might mean smooth etc. The only common theme is that the author didn't bother explaining it in more detail.






                    share|cite|improve this answer









                    $endgroup$



                    It might mean continuous, it might mean differentiable, it might mean smooth etc. The only common theme is that the author didn't bother explaining it in more detail.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Apr 22 '14 at 12:09









                    Edvard FagerholmEdvard Fagerholm

                    67839




                    67839























                        6












                        $begingroup$

                        Comment converted to answer:



                        I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."






                        share|cite|improve this answer









                        $endgroup$


















                          6












                          $begingroup$

                          Comment converted to answer:



                          I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."






                          share|cite|improve this answer









                          $endgroup$
















                            6












                            6








                            6





                            $begingroup$

                            Comment converted to answer:



                            I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."






                            share|cite|improve this answer









                            $endgroup$



                            Comment converted to answer:



                            I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Apr 22 '14 at 18:16









                            apnortonapnorton

                            15.2k33796




                            15.2k33796






























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