Properties of bounded linear operators between two normed spaces












0















Let $T$ be a linear operator between two normed spaces $X$ and $Y$. Show that





  • a) If $T$ is not bounded then for $varepsilon>0$ , $sup Vert TxVert = infty$ where $Vert x Vert < varepsilon$.

  • b) If ${T(x_{n})}$ is bounded for any sequence ${x_{n}}$ such that $x_nto 0$, then $T$ is bounded.


Please help me if you have a good idea for my question. I do not understand this concept. Thank you.










share|cite|improve this question
























  • @mathworker21 it's not nice to say that. The question is perfectly legit.
    – John_Wick
    Nov 24 at 14:15










  • @John_Wick i think you could have guessed he and I had a dialogue and then he deleted his comments
    – mathworker21
    Nov 24 at 14:16










  • Never mind. I apologize if I have offended you.
    – John_Wick
    Nov 24 at 14:17
















0















Let $T$ be a linear operator between two normed spaces $X$ and $Y$. Show that





  • a) If $T$ is not bounded then for $varepsilon>0$ , $sup Vert TxVert = infty$ where $Vert x Vert < varepsilon$.

  • b) If ${T(x_{n})}$ is bounded for any sequence ${x_{n}}$ such that $x_nto 0$, then $T$ is bounded.


Please help me if you have a good idea for my question. I do not understand this concept. Thank you.










share|cite|improve this question
























  • @mathworker21 it's not nice to say that. The question is perfectly legit.
    – John_Wick
    Nov 24 at 14:15










  • @John_Wick i think you could have guessed he and I had a dialogue and then he deleted his comments
    – mathworker21
    Nov 24 at 14:16










  • Never mind. I apologize if I have offended you.
    – John_Wick
    Nov 24 at 14:17














0












0








0








Let $T$ be a linear operator between two normed spaces $X$ and $Y$. Show that





  • a) If $T$ is not bounded then for $varepsilon>0$ , $sup Vert TxVert = infty$ where $Vert x Vert < varepsilon$.

  • b) If ${T(x_{n})}$ is bounded for any sequence ${x_{n}}$ such that $x_nto 0$, then $T$ is bounded.


Please help me if you have a good idea for my question. I do not understand this concept. Thank you.










share|cite|improve this question
















Let $T$ be a linear operator between two normed spaces $X$ and $Y$. Show that





  • a) If $T$ is not bounded then for $varepsilon>0$ , $sup Vert TxVert = infty$ where $Vert x Vert < varepsilon$.

  • b) If ${T(x_{n})}$ is bounded for any sequence ${x_{n}}$ such that $x_nto 0$, then $T$ is bounded.


Please help me if you have a good idea for my question. I do not understand this concept. Thank you.







functional-analysis linear-transformations normed-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 24 at 21:46









user587192

1,488112




1,488112










asked Nov 24 at 13:17









ezgialtinbasak

15




15












  • @mathworker21 it's not nice to say that. The question is perfectly legit.
    – John_Wick
    Nov 24 at 14:15










  • @John_Wick i think you could have guessed he and I had a dialogue and then he deleted his comments
    – mathworker21
    Nov 24 at 14:16










  • Never mind. I apologize if I have offended you.
    – John_Wick
    Nov 24 at 14:17


















  • @mathworker21 it's not nice to say that. The question is perfectly legit.
    – John_Wick
    Nov 24 at 14:15










  • @John_Wick i think you could have guessed he and I had a dialogue and then he deleted his comments
    – mathworker21
    Nov 24 at 14:16










  • Never mind. I apologize if I have offended you.
    – John_Wick
    Nov 24 at 14:17
















@mathworker21 it's not nice to say that. The question is perfectly legit.
– John_Wick
Nov 24 at 14:15




@mathworker21 it's not nice to say that. The question is perfectly legit.
– John_Wick
Nov 24 at 14:15












@John_Wick i think you could have guessed he and I had a dialogue and then he deleted his comments
– mathworker21
Nov 24 at 14:16




@John_Wick i think you could have guessed he and I had a dialogue and then he deleted his comments
– mathworker21
Nov 24 at 14:16












Never mind. I apologize if I have offended you.
– John_Wick
Nov 24 at 14:17




Never mind. I apologize if I have offended you.
– John_Wick
Nov 24 at 14:17










1 Answer
1






active

oldest

votes


















0














(a) If $sup_{|x|<epsilon} |Tx|=M<infty $ then $sup_{xin X,|x|>0}frac{|Tx|}{|x|}=sup_{xin X,|x|>0}|T(epsilon x/|x|)|/epsilonleq sup_{|x|<epsilon} |Tx|/epsilon=M/epsilon<infty. $ Hence $sup_{|x|<epsilon} |Tx|<infty$ implies $T$ is bounded.



(b) Let $T$ is unbounded. So, by (a) for any $ninmathbb{N}$, $sup_{|x|<1/n}=infty$. So there exists an $x_n$ with $|x_n|<1/n$ such that $|T(x_n)|geq n$. Hence $T$ unbounded implies there is a sequence $(x_n)$ converging to zero such that $|T(x_n)|rightarrow infty.$






share|cite|improve this answer























  • you could be dividing by 0 at one step in part a
    – mathworker21
    Nov 24 at 13:49












  • Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
    – John_Wick
    Nov 24 at 14:13










  • Thank you John really.
    – ezgialtinbasak
    Nov 24 at 14:22











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
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3011546%2fproperties-of-bounded-linear-operators-between-two-normed-spaces%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









0














(a) If $sup_{|x|<epsilon} |Tx|=M<infty $ then $sup_{xin X,|x|>0}frac{|Tx|}{|x|}=sup_{xin X,|x|>0}|T(epsilon x/|x|)|/epsilonleq sup_{|x|<epsilon} |Tx|/epsilon=M/epsilon<infty. $ Hence $sup_{|x|<epsilon} |Tx|<infty$ implies $T$ is bounded.



(b) Let $T$ is unbounded. So, by (a) for any $ninmathbb{N}$, $sup_{|x|<1/n}=infty$. So there exists an $x_n$ with $|x_n|<1/n$ such that $|T(x_n)|geq n$. Hence $T$ unbounded implies there is a sequence $(x_n)$ converging to zero such that $|T(x_n)|rightarrow infty.$






share|cite|improve this answer























  • you could be dividing by 0 at one step in part a
    – mathworker21
    Nov 24 at 13:49












  • Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
    – John_Wick
    Nov 24 at 14:13










  • Thank you John really.
    – ezgialtinbasak
    Nov 24 at 14:22
















0














(a) If $sup_{|x|<epsilon} |Tx|=M<infty $ then $sup_{xin X,|x|>0}frac{|Tx|}{|x|}=sup_{xin X,|x|>0}|T(epsilon x/|x|)|/epsilonleq sup_{|x|<epsilon} |Tx|/epsilon=M/epsilon<infty. $ Hence $sup_{|x|<epsilon} |Tx|<infty$ implies $T$ is bounded.



(b) Let $T$ is unbounded. So, by (a) for any $ninmathbb{N}$, $sup_{|x|<1/n}=infty$. So there exists an $x_n$ with $|x_n|<1/n$ such that $|T(x_n)|geq n$. Hence $T$ unbounded implies there is a sequence $(x_n)$ converging to zero such that $|T(x_n)|rightarrow infty.$






share|cite|improve this answer























  • you could be dividing by 0 at one step in part a
    – mathworker21
    Nov 24 at 13:49












  • Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
    – John_Wick
    Nov 24 at 14:13










  • Thank you John really.
    – ezgialtinbasak
    Nov 24 at 14:22














0












0








0






(a) If $sup_{|x|<epsilon} |Tx|=M<infty $ then $sup_{xin X,|x|>0}frac{|Tx|}{|x|}=sup_{xin X,|x|>0}|T(epsilon x/|x|)|/epsilonleq sup_{|x|<epsilon} |Tx|/epsilon=M/epsilon<infty. $ Hence $sup_{|x|<epsilon} |Tx|<infty$ implies $T$ is bounded.



(b) Let $T$ is unbounded. So, by (a) for any $ninmathbb{N}$, $sup_{|x|<1/n}=infty$. So there exists an $x_n$ with $|x_n|<1/n$ such that $|T(x_n)|geq n$. Hence $T$ unbounded implies there is a sequence $(x_n)$ converging to zero such that $|T(x_n)|rightarrow infty.$






share|cite|improve this answer














(a) If $sup_{|x|<epsilon} |Tx|=M<infty $ then $sup_{xin X,|x|>0}frac{|Tx|}{|x|}=sup_{xin X,|x|>0}|T(epsilon x/|x|)|/epsilonleq sup_{|x|<epsilon} |Tx|/epsilon=M/epsilon<infty. $ Hence $sup_{|x|<epsilon} |Tx|<infty$ implies $T$ is bounded.



(b) Let $T$ is unbounded. So, by (a) for any $ninmathbb{N}$, $sup_{|x|<1/n}=infty$. So there exists an $x_n$ with $|x_n|<1/n$ such that $|T(x_n)|geq n$. Hence $T$ unbounded implies there is a sequence $(x_n)$ converging to zero such that $|T(x_n)|rightarrow infty.$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 24 at 14:14

























answered Nov 24 at 13:44









John_Wick

1,314111




1,314111












  • you could be dividing by 0 at one step in part a
    – mathworker21
    Nov 24 at 13:49












  • Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
    – John_Wick
    Nov 24 at 14:13










  • Thank you John really.
    – ezgialtinbasak
    Nov 24 at 14:22


















  • you could be dividing by 0 at one step in part a
    – mathworker21
    Nov 24 at 13:49












  • Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
    – John_Wick
    Nov 24 at 14:13










  • Thank you John really.
    – ezgialtinbasak
    Nov 24 at 14:22
















you could be dividing by 0 at one step in part a
– mathworker21
Nov 24 at 13:49






you could be dividing by 0 at one step in part a
– mathworker21
Nov 24 at 13:49














Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
– John_Wick
Nov 24 at 14:13




Oh, for that take $x$ such that $|x|>0.$ I am going to edit my answer.
– John_Wick
Nov 24 at 14:13












Thank you John really.
– ezgialtinbasak
Nov 24 at 14:22




Thank you John really.
– ezgialtinbasak
Nov 24 at 14:22


















draft saved

draft discarded




















































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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3011546%2fproperties-of-bounded-linear-operators-between-two-normed-spaces%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Ellipse (mathématiques)

Quarter-circle Tiles

Mont Emei