How is $C_0^k(mathbb R)$ defined?
I'm sorry for asking such a short question, but I see the space $C_0^k(mathbb R)$ everywhere being used without a rigorous definition.
$C_0(mathbb R)$ is the space of continuous functions on $mathbb R$ vanishing at infinity. The question is, is $C_0^k(mathbb R):=C_0(mathbb R)cap C^k(mathbb R)$ or is $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$?
Those spaces shouldn't coincide.
analysis derivatives
asked Nov 24 at 13:01
0xbadf00d
1,81441430
add a comment |
I'm sorry for asking such a short question, but I see the space $C_0^k(mathbb R)$ everywhere being used without a rigorous definition.
$C_0(mathbb R)$ is the space of continuous functions on $mathbb R$ vanishing at infinity. The question is, is $C_0^k(mathbb R):=C_0(mathbb R)cap C^k(mathbb R)$ or is $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$?
Those spaces shouldn't coincide.
analysis derivatives
asked Nov 24 at 13:01
0xbadf00d
1,81441430
add a comment |
I'm sorry for asking such a short question, but I see the space $C_0^k(mathbb R)$ everywhere being used without a rigorous definition.
$C_0(mathbb R)$ is the space of continuous functions on $mathbb R$ vanishing at infinity. The question is, is $C_0^k(mathbb R):=C_0(mathbb R)cap C^k(mathbb R)$ or is $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$?
Those spaces shouldn't coincide.
analysis derivatives
asked Nov 24 at 13:01
0xbadf00d
1,81441430
I'm sorry for asking such a short question, but I see the space $C_0^k(mathbb R)$ everywhere being used without a rigorous definition.
$C_0(mathbb R)$ is the space of continuous functions on $mathbb R$ vanishing at infinity. The question is, is $C_0^k(mathbb R):=C_0(mathbb R)cap C^k(mathbb R)$ or is $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$?
Those spaces shouldn't coincide.
analysis derivatives
analysis derivatives
asked Nov 24 at 13:01
0xbadf00d
1,81441430
asked Nov 24 at 13:01
0xbadf00d
1,81441430
asked Nov 24 at 13:01
0xbadf00d
1,81441430
asked Nov 24 at 13:01
0xbadf00d
1,81441430
asked Nov 24 at 13:01
0xbadf00d
1,81441430
1,81441430
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Correct definition is that $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$.
If you write $C_c(Bbb R)$ set of all continuous function on $Bbb R$ with compact support then $C_c^k(mathbb R):=C_c(mathbb R)cap C^k(mathbb R)$.
answered Nov 24 at 13:39
UserS
1,5371112
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
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%2f3011524%2fhow-is-c-0k-mathbb-r-defined%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Correct definition is that $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$.
If you write $C_c(Bbb R)$ set of all continuous function on $Bbb R$ with compact support then $C_c^k(mathbb R):=C_c(mathbb R)cap C^k(mathbb R)$.
answered Nov 24 at 13:39
UserS
1,5371112
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
add a comment |
Correct definition is that $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$.
If you write $C_c(Bbb R)$ set of all continuous function on $Bbb R$ with compact support then $C_c^k(mathbb R):=C_c(mathbb R)cap C^k(mathbb R)$.
answered Nov 24 at 13:39
UserS
1,5371112
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
add a comment |
Correct definition is that $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$.
If you write $C_c(Bbb R)$ set of all continuous function on $Bbb R$ with compact support then $C_c^k(mathbb R):=C_c(mathbb R)cap C^k(mathbb R)$.
answered Nov 24 at 13:39
UserS
1,5371112
Correct definition is that $C_0^k(mathbb R):=left{fin C^k(mathbb R):f^{(i)}in C_0(mathbb R)text{ for all }iinleft{0,ldots,kright}right}$.
If you write $C_c(Bbb R)$ set of all continuous function on $Bbb R$ with compact support then $C_c^k(mathbb R):=C_c(mathbb R)cap C^k(mathbb R)$.
answered Nov 24 at 13:39
UserS
1,5371112
answered Nov 24 at 13:39
UserS
1,5371112
answered Nov 24 at 13:39
UserS
1,5371112
answered Nov 24 at 13:39
UserS
1,5371112
1,5371112
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
add a comment |
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
I follow the notation of the book "PSEUDO DIFFERENTIAL OPERATORS and MARKOV PROCESSES" written by N. Jacob. See page xiv for notation.
– UserS
Nov 24 at 13:42
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
$C_0(mathbb R)$ can be defined to be the closure of $C_c^infty(mathbb R)$ in $C_b(mathbb R)$ (bounded continuous functions on $mathbb R$. Is in the same way $C^k_0(mathbb R)$ the closure of $C_c^k(mathbb R)$ in $left{fin C^k(mathbb R):f^{(i)}text{ is bounded for all }iinleft{0,ldots,kright}right}$ equipped with $sum_{i=0}^kleft|f^{(i)}right|_infty$?
– 0xbadf00d
Nov 24 at 17:31
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%2f3011524%2fhow-is-c-0k-mathbb-r-defined%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
