Bruhat-Schwartz functions on pro-finite groups
$begingroup$
Let $X$ be a profinite topological group equipped with its Haar measure $mu$ of total volume $1$, and $S(X,mathbf{C})$ be the complex vector space of those continuous functions $f : Xto mathbf{C}$ that are locally constant on $X$.
Let $C(X,mathbf{C})$ be the complex vector space of all continuous functions $f : Xtomathbf{C}$.
Is $S(X,mathbf{C})$ dense in $C(X,mathbf{C})$ with respect to the $L^2$-norm:
$$|f|_2 := int_X|f|^2mu ?$$
real-analysis functional-analysis representation-theory algebraic-number-theory topological-groups
asked Dec 20 '18 at 14:04
user628163user628163
12
$endgroup$
add a comment |
$begingroup$
Let $X$ be a profinite topological group equipped with its Haar measure $mu$ of total volume $1$, and $S(X,mathbf{C})$ be the complex vector space of those continuous functions $f : Xto mathbf{C}$ that are locally constant on $X$.
Let $C(X,mathbf{C})$ be the complex vector space of all continuous functions $f : Xtomathbf{C}$.
Is $S(X,mathbf{C})$ dense in $C(X,mathbf{C})$ with respect to the $L^2$-norm:
$$|f|_2 := int_X|f|^2mu ?$$
real-analysis functional-analysis representation-theory algebraic-number-theory topological-groups
asked Dec 20 '18 at 14:04
user628163user628163
12
$endgroup$
add a comment |
$begingroup$
Let $X$ be a profinite topological group equipped with its Haar measure $mu$ of total volume $1$, and $S(X,mathbf{C})$ be the complex vector space of those continuous functions $f : Xto mathbf{C}$ that are locally constant on $X$.
Let $C(X,mathbf{C})$ be the complex vector space of all continuous functions $f : Xtomathbf{C}$.
Is $S(X,mathbf{C})$ dense in $C(X,mathbf{C})$ with respect to the $L^2$-norm:
$$|f|_2 := int_X|f|^2mu ?$$
real-analysis functional-analysis representation-theory algebraic-number-theory topological-groups
asked Dec 20 '18 at 14:04
user628163user628163
12
$endgroup$
Let $X$ be a profinite topological group equipped with its Haar measure $mu$ of total volume $1$, and $S(X,mathbf{C})$ be the complex vector space of those continuous functions $f : Xto mathbf{C}$ that are locally constant on $X$.
Let $C(X,mathbf{C})$ be the complex vector space of all continuous functions $f : Xtomathbf{C}$.
Is $S(X,mathbf{C})$ dense in $C(X,mathbf{C})$ with respect to the $L^2$-norm:
$$|f|_2 := int_X|f|^2mu ?$$
real-analysis functional-analysis representation-theory algebraic-number-theory topological-groups
real-analysis functional-analysis representation-theory algebraic-number-theory topological-groups
asked Dec 20 '18 at 14:04
user628163user628163
12
asked Dec 20 '18 at 14:04
user628163user628163
12
edited Dec 20 '18 at 14:22
user628163
asked Dec 20 '18 at 14:04
user628163user628163
12
asked Dec 20 '18 at 14:04
user628163user628163
12
asked Dec 20 '18 at 14:04
user628163user628163
12
12
add a comment |
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%2f3047569%2fbruhat-schwartz-functions-on-pro-finite-groups%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%2f3047569%2fbruhat-schwartz-functions-on-pro-finite-groups%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
