Mixing
About Locally Compact Groups and Analysis
$begingroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
$endgroup$
add a comment |
$begingroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
$endgroup$
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
add a comment |
$begingroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
$endgroup$
This question was asked in my previous Ph.D. qualifying for Analysis and I couldn't solve it. I have no clue on how to proceed.
Let $G$ be a locally compact group and let $f in C_c(G)$ where $C_c(G)$ is the set of all continuous functions on $G$ with compact supports. Then $forall epsilon > 0$, there is an open neighborhood $U$ of the identity such that whenever $y in xU$, it follows that $|f(x) - f(y)| < epsilon$.
What is the identity? what is a locally compact group? and what they mean for $xU$ ? a very odd problem for an analysis qualifying.
real-analysis measure-theory continuity locally-compact-groups
real-analysis measure-theory continuity locally-compact-groups
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
edited Jan 10 at 17:08
David C. Ullrich
60.5k43994
60.5k43994
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
asked Jan 10 at 12:06
Richard ClareRichard Clare
1,066314
1,066314
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
add a comment |
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
1
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09
add a comment |
0
active
oldest
votes
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%2f3068559%2fabout-locally-compact-groups-and-analysis%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3068559%2fabout-locally-compact-groups-and-analysis%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
1
$begingroup$
All the relevant definitions can be found on, say, Wikipedia. In the end, the problem boils down to showing that every continuous function on a compact space is uniformly continuous.
$endgroup$
– MaoWao
Jan 10 at 12:19
$begingroup$
Wow... that was a low blow. Now I see it. I know how to prove that one.
$endgroup$
– Richard Clare
Jan 10 at 12:31
$begingroup$
Probably you realize this, but note that $C_c(G)$ is the space of continuous functions with compact support...
$endgroup$
– David C. Ullrich
Jan 10 at 17:09