“nice functions”
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
asked Apr 22 '14 at 12:02
bjk1806
160112
add a comment |
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
asked Apr 22 '14 at 12:02
bjk1806
160112
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.
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11
7
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
– apnorton
Apr 22 '14 at 14:01
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
– Mario Carneiro
Apr 22 '14 at 16:52
@MarioCarneiro Done.
– apnorton
Apr 22 '14 at 18:16
add a comment |
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
asked Apr 22 '14 at 12:02
bjk1806
160112
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
asked Apr 22 '14 at 12:02
bjk1806
160112
asked Apr 22 '14 at 12:02
bjk1806
160112
asked Apr 22 '14 at 12:02
bjk1806
160112
asked Apr 22 '14 at 12:02
bjk1806
160112
asked Apr 22 '14 at 12:02
bjk1806
160112
160112
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.
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11
7
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
– apnorton
Apr 22 '14 at 14:01
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
– Mario Carneiro
Apr 22 '14 at 16:52
@MarioCarneiro Done.
– apnorton
Apr 22 '14 at 18:16
add a comment |
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.
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11
7
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
– apnorton
Apr 22 '14 at 14:01
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
– Mario Carneiro
Apr 22 '14 at 16:52
@MarioCarneiro Done.
– apnorton
Apr 22 '14 at 18:16
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.
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11
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.
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11
7
7
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
– apnorton
Apr 22 '14 at 14:01
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
– apnorton
Apr 22 '14 at 14:01
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
– Mario Carneiro
Apr 22 '14 at 16:52
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
– Mario Carneiro
Apr 22 '14 at 16:52
@MarioCarneiro Done.
– apnorton
Apr 22 '14 at 18:16
@MarioCarneiro Done.
– apnorton
Apr 22 '14 at 18:16
add a comment |
3 Answers
3
active
oldest
votes
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.
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
add a comment |
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.
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
add a comment |
Comment converted to answer:
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f764354%2fnice-functions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
add a comment |
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.
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
add a comment |
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.
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
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.
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
answered Apr 22 '14 at 12:19
Elchanan Solomon
21.7k54277
21.7k54277
add a comment |
add a comment |
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.
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
add a comment |
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.
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
add a comment |
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.
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
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.
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
answered Apr 22 '14 at 12:09
Edvard Fagerholm
67839
67839
add a comment |
add a comment |
Comment converted to answer:
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
add a comment |
Comment converted to answer:
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
add a comment |
Comment converted to answer:
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
Comment converted to answer:
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
answered Apr 22 '14 at 18:16
apnorton
15.1k33696
15.1k33696
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f764354%2fnice-functions%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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.
– Stephen Montgomery-Smith
Apr 22 '14 at 12:11
7
I read "nice" or "well-behaved" as "satisfying all requirements for the appropriate theorems I am using."
– apnorton
Apr 22 '14 at 14:01
@anorton You should make that an answer, so I can vote for it (that's the most correct answer I see right now)
– Mario Carneiro
Apr 22 '14 at 16:52
@MarioCarneiro Done.
– apnorton
Apr 22 '14 at 18:16