$U={xin mathbb{R^n} :g_i(x)leq 0 : text{for} quad i=1,…,m }$ is closed
$begingroup$
Let $U={xin mathbb{R^n} :g_i(x)leq 0 : forall i=1,...,m }$ $subset$ $mathbb{R^n}$.
with $(g_i)_{1leq i leq n}$ are convex functions from $mathbb{R^n}$ in $mathbb{R}$.
we want to prove that $U$ is a closed set.
general-topology convex-analysis
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
$endgroup$
add a comment |
$begingroup$
Let $U={xin mathbb{R^n} :g_i(x)leq 0 : forall i=1,...,m }$ $subset$ $mathbb{R^n}$.
with $(g_i)_{1leq i leq n}$ are convex functions from $mathbb{R^n}$ in $mathbb{R}$.
we want to prove that $U$ is a closed set.
general-topology convex-analysis
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
$endgroup$
$begingroup$
Should it be $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$?
$endgroup$
– TrostAft
Dec 19 '18 at 7:23
add a comment |
$begingroup$
Let $U={xin mathbb{R^n} :g_i(x)leq 0 : forall i=1,...,m }$ $subset$ $mathbb{R^n}$.
with $(g_i)_{1leq i leq n}$ are convex functions from $mathbb{R^n}$ in $mathbb{R}$.
we want to prove that $U$ is a closed set.
general-topology convex-analysis
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
$endgroup$
Let $U={xin mathbb{R^n} :g_i(x)leq 0 : forall i=1,...,m }$ $subset$ $mathbb{R^n}$.
with $(g_i)_{1leq i leq n}$ are convex functions from $mathbb{R^n}$ in $mathbb{R}$.
we want to prove that $U$ is a closed set.
general-topology convex-analysis
general-topology convex-analysis
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
edited Dec 19 '18 at 7:46
Hossien Sahebjame
817
817
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
asked Dec 19 '18 at 7:21
Abderrahmane OuachouachAbderrahmane Ouachouach
61
61
$begingroup$
Should it be $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$?
$endgroup$
– TrostAft
Dec 19 '18 at 7:23
add a comment |
$begingroup$
Should it be $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$?
$endgroup$
– TrostAft
Dec 19 '18 at 7:23
$begingroup$
Should it be $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$?
$endgroup$
– TrostAft
Dec 19 '18 at 7:23
$begingroup$
Should it be $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$?
$endgroup$
– TrostAft
Dec 19 '18 at 7:23
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Any convex function from $mathbb R^{n}$ to $mathbb R$ is continuous. It follows that $U$ is the intersection of a $m$ closed sets, hence closed.
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
$endgroup$
add a comment |
$begingroup$
I think we have $g=(g_1,...,g_m): mathbb R^n to mathbb R^m$ and $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$ with each $g_i$ convex.
Let $U_i := {x in mathbb{R}^n : g_i(x) leq 0}$.
Since $g_i$ is continuous, we have that $U_i$ is closed. Hence $U= bigcap_{i=1}^n U_i$ is closed.
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
$endgroup$
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%2f3046129%2fu-x-in-mathbbrn-g-ix-leq-0-textfor-quad-i-1-m-is-closed%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Any convex function from $mathbb R^{n}$ to $mathbb R$ is continuous. It follows that $U$ is the intersection of a $m$ closed sets, hence closed.
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
$endgroup$
add a comment |
$begingroup$
Any convex function from $mathbb R^{n}$ to $mathbb R$ is continuous. It follows that $U$ is the intersection of a $m$ closed sets, hence closed.
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
$endgroup$
add a comment |
$begingroup$
Any convex function from $mathbb R^{n}$ to $mathbb R$ is continuous. It follows that $U$ is the intersection of a $m$ closed sets, hence closed.
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
$endgroup$
Any convex function from $mathbb R^{n}$ to $mathbb R$ is continuous. It follows that $U$ is the intersection of a $m$ closed sets, hence closed.
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
answered Dec 19 '18 at 7:29
Kavi Rama MurthyKavi Rama Murthy
56k42158
56k42158
add a comment |
add a comment |
$begingroup$
I think we have $g=(g_1,...,g_m): mathbb R^n to mathbb R^m$ and $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$ with each $g_i$ convex.
Let $U_i := {x in mathbb{R}^n : g_i(x) leq 0}$.
Since $g_i$ is continuous, we have that $U_i$ is closed. Hence $U= bigcap_{i=1}^n U_i$ is closed.
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
$endgroup$
add a comment |
$begingroup$
I think we have $g=(g_1,...,g_m): mathbb R^n to mathbb R^m$ and $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$ with each $g_i$ convex.
Let $U_i := {x in mathbb{R}^n : g_i(x) leq 0}$.
Since $g_i$ is continuous, we have that $U_i$ is closed. Hence $U= bigcap_{i=1}^n U_i$ is closed.
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
$endgroup$
add a comment |
$begingroup$
I think we have $g=(g_1,...,g_m): mathbb R^n to mathbb R^m$ and $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$ with each $g_i$ convex.
Let $U_i := {x in mathbb{R}^n : g_i(x) leq 0}$.
Since $g_i$ is continuous, we have that $U_i$ is closed. Hence $U= bigcap_{i=1}^n U_i$ is closed.
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
$endgroup$
I think we have $g=(g_1,...,g_m): mathbb R^n to mathbb R^m$ and $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$ with each $g_i$ convex.
Let $U_i := {x in mathbb{R}^n : g_i(x) leq 0}$.
Since $g_i$ is continuous, we have that $U_i$ is closed. Hence $U= bigcap_{i=1}^n U_i$ is closed.
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
answered Dec 19 '18 at 7:30
FredFred
45.2k1847
45.2k1847
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.
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%2f3046129%2fu-x-in-mathbbrn-g-ix-leq-0-textfor-quad-i-1-m-is-closed%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
$begingroup$
Should it be $U = {x in mathbb{R}^n : g_i(x) leq 0, forall i = 1, dots, m}$?
$endgroup$
– TrostAft
Dec 19 '18 at 7:23