Prove that $Vertcdot Vert^2:Xto Bbb{R},$ where $X$ is a vector space, is convex












6












$begingroup$


Let $X$ be a vector space. I was able to prove that $Vertcdot Vert:Xto Bbb{R},$ is a convex function, i.e., for all $x,yin X$ and $lambda in [0,1],$



begin{align} Vert lambda x+(1-lambda)y Vert leq lambda Vert xVert+(1-lambda)Vert y Vertend{align}



Now, I want to prove that $Vertcdot Vert^2:Xto Bbb{R},$ where $X$ is a vector space, is convex. So, here's what I've done!



MY WORK



begin{align} Vert lambda x+(1-lambda)y Vert^2 leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2,;;text{for all};; x,yin X;; text{and};; lambda in [0,1].end{align}



So, any help please on how to proceed?






functional-analysis analysis convex-analysis norm normed-spaces







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    do you want to specifically know if $|cdot|^2$ is convex? then you should make this more clear, also in the title
    $endgroup$
    – supinf
    Nov 30 '18 at 10:40










  • $begingroup$
    @supinf: I made some edits!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:16










  • $begingroup$
    Does your vector space also provide inner product?
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:54
















6












$begingroup$


Let $X$ be a vector space. I was able to prove that $Vertcdot Vert:Xto Bbb{R},$ is a convex function, i.e., for all $x,yin X$ and $lambda in [0,1],$



begin{align} Vert lambda x+(1-lambda)y Vert leq lambda Vert xVert+(1-lambda)Vert y Vertend{align}



Now, I want to prove that $Vertcdot Vert^2:Xto Bbb{R},$ where $X$ is a vector space, is convex. So, here's what I've done!



MY WORK



begin{align} Vert lambda x+(1-lambda)y Vert^2 leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2,;;text{for all};; x,yin X;; text{and};; lambda in [0,1].end{align}



So, any help please on how to proceed?






functional-analysis analysis convex-analysis norm normed-spaces







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    do you want to specifically know if $|cdot|^2$ is convex? then you should make this more clear, also in the title
    $endgroup$
    – supinf
    Nov 30 '18 at 10:40










  • $begingroup$
    @supinf: I made some edits!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:16










  • $begingroup$
    Does your vector space also provide inner product?
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:54














6












6








6


1



$begingroup$


Let $X$ be a vector space. I was able to prove that $Vertcdot Vert:Xto Bbb{R},$ is a convex function, i.e., for all $x,yin X$ and $lambda in [0,1],$



begin{align} Vert lambda x+(1-lambda)y Vert leq lambda Vert xVert+(1-lambda)Vert y Vertend{align}



Now, I want to prove that $Vertcdot Vert^2:Xto Bbb{R},$ where $X$ is a vector space, is convex. So, here's what I've done!



MY WORK



begin{align} Vert lambda x+(1-lambda)y Vert^2 leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2,;;text{for all};; x,yin X;; text{and};; lambda in [0,1].end{align}



So, any help please on how to proceed?






functional-analysis analysis convex-analysis norm normed-spaces







share|cite|improve this question











$endgroup$




Let $X$ be a vector space. I was able to prove that $Vertcdot Vert:Xto Bbb{R},$ is a convex function, i.e., for all $x,yin X$ and $lambda in [0,1],$



begin{align} Vert lambda x+(1-lambda)y Vert leq lambda Vert xVert+(1-lambda)Vert y Vertend{align}



Now, I want to prove that $Vertcdot Vert^2:Xto Bbb{R},$ where $X$ is a vector space, is convex. So, here's what I've done!



MY WORK



begin{align} Vert lambda x+(1-lambda)y Vert^2 leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2,;;text{for all};; x,yin X;; text{and};; lambda in [0,1].end{align}



So, any help please on how to proceed?






functional-analysis analysis convex-analysis norm normed-spaces




functional-analysis analysis convex-analysis norm normed-spaces






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Nov 30 '18 at 11:25







Omojola Micheal

















asked Nov 30 '18 at 10:30









Omojola MichealOmojola Micheal

1,753324




1,753324








  • 3




    $begingroup$
    do you want to specifically know if $|cdot|^2$ is convex? then you should make this more clear, also in the title
    $endgroup$
    – supinf
    Nov 30 '18 at 10:40










  • $begingroup$
    @supinf: I made some edits!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:16










  • $begingroup$
    Does your vector space also provide inner product?
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:54














  • 3




    $begingroup$
    do you want to specifically know if $|cdot|^2$ is convex? then you should make this more clear, also in the title
    $endgroup$
    – supinf
    Nov 30 '18 at 10:40










  • $begingroup$
    @supinf: I made some edits!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:16










  • $begingroup$
    Does your vector space also provide inner product?
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:54








3




3




$begingroup$
do you want to specifically know if $|cdot|^2$ is convex? then you should make this more clear, also in the title
$endgroup$
– supinf
Nov 30 '18 at 10:40




$begingroup$
do you want to specifically know if $|cdot|^2$ is convex? then you should make this more clear, also in the title
$endgroup$
– supinf
Nov 30 '18 at 10:40












$begingroup$
@supinf: I made some edits!
$endgroup$
– Omojola Micheal
Nov 30 '18 at 11:16




$begingroup$
@supinf: I made some edits!
$endgroup$
– Omojola Micheal
Nov 30 '18 at 11:16












$begingroup$
Does your vector space also provide inner product?
$endgroup$
– Mostafa Ayaz
Nov 30 '18 at 11:54




$begingroup$
Does your vector space also provide inner product?
$endgroup$
– Mostafa Ayaz
Nov 30 '18 at 11:54










3 Answers
3






active

oldest

votes


















2












$begingroup$

In general if $f$ is a convex function and $g$ is a convex nondecreasing function then the composition $g circ f$ is a convex function. Let $f(cdot)=|cdot |$ which maps to $mathbb{R}_{geq 0}$ and let $g(x)=x^2$ which is a nondecreasing convex function on $mathbb{R}_{geq 0}$. If follows that $g circ f (cdot)=| cdot |^2$ is a convex function.



See The composition of two convex functions is convex for the original claim.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is good! I like it!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31



















4












$begingroup$

Just solved and thought to share it for the sake of future readers.
begin{align} Vert lambda x+(1-lambda)y Vert^2 &leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2\ &leq lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2\ &= lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2 -lambdaVert xVert^2 -(1-lambda)Vert yVert^2\&quad+lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{adding and substracting};lambdaVert xVert^2 +(1-lambda)Vert yVert^2 \ &= -lambda (1-lambda)left(Vert xVert-Vert yVertright)^2+lambdaVert xVert^2 +(1-lambda)Vert yVert^2\ &leq lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{since};-lambda (1-lambda)left(Vert xVert-Vert yVertright)^2leq 0.end{align}
Hence, $VertcdotVert^2$ is a convex function.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Nice! (+1)......
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:53



















1












$begingroup$

Define $p=lambda x$ and $q=(1-lambda)y$, therefore we need to show that $$||p+q||^2le (||p||+||q||)^2$$which reduces to $$pcdot qle ||p||cdot ||q||$$which is the same famous Cauchy-Schwartz inequality. Therefore $||.||^2$ is convex.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    That's fine too!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31










  • $begingroup$
    Thank you. Good luck!
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:33










  • $begingroup$
    This only works if the norm comes from a scalar product, however.
    $endgroup$
    – Giuseppe Negro
    Nov 30 '18 at 11:50











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%2f3019935%2fprove-that-vert-cdot-vert2x-to-bbbr-where-x-is-a-vector-space-is-co%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























3 Answers
3






active

oldest

votes








3 Answers
3






active

oldest

votes









active

oldest

votes






active

oldest

votes









2












$begingroup$

In general if $f$ is a convex function and $g$ is a convex nondecreasing function then the composition $g circ f$ is a convex function. Let $f(cdot)=|cdot |$ which maps to $mathbb{R}_{geq 0}$ and let $g(x)=x^2$ which is a nondecreasing convex function on $mathbb{R}_{geq 0}$. If follows that $g circ f (cdot)=| cdot |^2$ is a convex function.



See The composition of two convex functions is convex for the original claim.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is good! I like it!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31
















2












$begingroup$

In general if $f$ is a convex function and $g$ is a convex nondecreasing function then the composition $g circ f$ is a convex function. Let $f(cdot)=|cdot |$ which maps to $mathbb{R}_{geq 0}$ and let $g(x)=x^2$ which is a nondecreasing convex function on $mathbb{R}_{geq 0}$. If follows that $g circ f (cdot)=| cdot |^2$ is a convex function.



See The composition of two convex functions is convex for the original claim.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    This is good! I like it!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31














2












2








2





$begingroup$

In general if $f$ is a convex function and $g$ is a convex nondecreasing function then the composition $g circ f$ is a convex function. Let $f(cdot)=|cdot |$ which maps to $mathbb{R}_{geq 0}$ and let $g(x)=x^2$ which is a nondecreasing convex function on $mathbb{R}_{geq 0}$. If follows that $g circ f (cdot)=| cdot |^2$ is a convex function.



See The composition of two convex functions is convex for the original claim.






share|cite|improve this answer









$endgroup$



In general if $f$ is a convex function and $g$ is a convex nondecreasing function then the composition $g circ f$ is a convex function. Let $f(cdot)=|cdot |$ which maps to $mathbb{R}_{geq 0}$ and let $g(x)=x^2$ which is a nondecreasing convex function on $mathbb{R}_{geq 0}$. If follows that $g circ f (cdot)=| cdot |^2$ is a convex function.



See The composition of two convex functions is convex for the original claim.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 30 '18 at 11:05









EricEric

2068




2068












  • $begingroup$
    This is good! I like it!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31


















  • $begingroup$
    This is good! I like it!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31
















$begingroup$
This is good! I like it!
$endgroup$
– Omojola Micheal
Nov 30 '18 at 11:31




$begingroup$
This is good! I like it!
$endgroup$
– Omojola Micheal
Nov 30 '18 at 11:31











4












$begingroup$

Just solved and thought to share it for the sake of future readers.
begin{align} Vert lambda x+(1-lambda)y Vert^2 &leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2\ &leq lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2\ &= lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2 -lambdaVert xVert^2 -(1-lambda)Vert yVert^2\&quad+lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{adding and substracting};lambdaVert xVert^2 +(1-lambda)Vert yVert^2 \ &= -lambda (1-lambda)left(Vert xVert-Vert yVertright)^2+lambdaVert xVert^2 +(1-lambda)Vert yVert^2\ &leq lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{since};-lambda (1-lambda)left(Vert xVert-Vert yVertright)^2leq 0.end{align}
Hence, $VertcdotVert^2$ is a convex function.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Nice! (+1)......
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:53
















4












$begingroup$

Just solved and thought to share it for the sake of future readers.
begin{align} Vert lambda x+(1-lambda)y Vert^2 &leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2\ &leq lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2\ &= lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2 -lambdaVert xVert^2 -(1-lambda)Vert yVert^2\&quad+lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{adding and substracting};lambdaVert xVert^2 +(1-lambda)Vert yVert^2 \ &= -lambda (1-lambda)left(Vert xVert-Vert yVertright)^2+lambdaVert xVert^2 +(1-lambda)Vert yVert^2\ &leq lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{since};-lambda (1-lambda)left(Vert xVert-Vert yVertright)^2leq 0.end{align}
Hence, $VertcdotVert^2$ is a convex function.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Nice! (+1)......
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:53














4












4








4





$begingroup$

Just solved and thought to share it for the sake of future readers.
begin{align} Vert lambda x+(1-lambda)y Vert^2 &leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2\ &leq lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2\ &= lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2 -lambdaVert xVert^2 -(1-lambda)Vert yVert^2\&quad+lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{adding and substracting};lambdaVert xVert^2 +(1-lambda)Vert yVert^2 \ &= -lambda (1-lambda)left(Vert xVert-Vert yVertright)^2+lambdaVert xVert^2 +(1-lambda)Vert yVert^2\ &leq lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{since};-lambda (1-lambda)left(Vert xVert-Vert yVertright)^2leq 0.end{align}
Hence, $VertcdotVert^2$ is a convex function.






share|cite|improve this answer











$endgroup$



Just solved and thought to share it for the sake of future readers.
begin{align} Vert lambda x+(1-lambda)y Vert^2 &leq left( lambda Vert xVert+(1-lambda)Vert y Vertright)^2\ &leq lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2\ &= lambda^2 Vert xVert^2+2lambda(1-lambda)Vert xVertVert yVert+ (1-lambda)^2Vert yVert^2 -lambdaVert xVert^2 -(1-lambda)Vert yVert^2\&quad+lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{adding and substracting};lambdaVert xVert^2 +(1-lambda)Vert yVert^2 \ &= -lambda (1-lambda)left(Vert xVert-Vert yVertright)^2+lambdaVert xVert^2 +(1-lambda)Vert yVert^2\ &leq lambdaVert xVert^2 +(1-lambda)Vert yVert^2,;;text{since};-lambda (1-lambda)left(Vert xVert-Vert yVertright)^2leq 0.end{align}
Hence, $VertcdotVert^2$ is a convex function.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Nov 30 '18 at 11:20

























answered Nov 30 '18 at 11:15









Omojola MichealOmojola Micheal

1,753324




1,753324












  • $begingroup$
    Nice! (+1)......
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:53


















  • $begingroup$
    Nice! (+1)......
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:53
















$begingroup$
Nice! (+1)......
$endgroup$
– Mostafa Ayaz
Nov 30 '18 at 11:53




$begingroup$
Nice! (+1)......
$endgroup$
– Mostafa Ayaz
Nov 30 '18 at 11:53











1












$begingroup$

Define $p=lambda x$ and $q=(1-lambda)y$, therefore we need to show that $$||p+q||^2le (||p||+||q||)^2$$which reduces to $$pcdot qle ||p||cdot ||q||$$which is the same famous Cauchy-Schwartz inequality. Therefore $||.||^2$ is convex.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    That's fine too!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31










  • $begingroup$
    Thank you. Good luck!
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:33










  • $begingroup$
    This only works if the norm comes from a scalar product, however.
    $endgroup$
    – Giuseppe Negro
    Nov 30 '18 at 11:50
















1












$begingroup$

Define $p=lambda x$ and $q=(1-lambda)y$, therefore we need to show that $$||p+q||^2le (||p||+||q||)^2$$which reduces to $$pcdot qle ||p||cdot ||q||$$which is the same famous Cauchy-Schwartz inequality. Therefore $||.||^2$ is convex.






share|cite|improve this answer









$endgroup$













  • $begingroup$
    That's fine too!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31










  • $begingroup$
    Thank you. Good luck!
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:33










  • $begingroup$
    This only works if the norm comes from a scalar product, however.
    $endgroup$
    – Giuseppe Negro
    Nov 30 '18 at 11:50














1












1








1





$begingroup$

Define $p=lambda x$ and $q=(1-lambda)y$, therefore we need to show that $$||p+q||^2le (||p||+||q||)^2$$which reduces to $$pcdot qle ||p||cdot ||q||$$which is the same famous Cauchy-Schwartz inequality. Therefore $||.||^2$ is convex.






share|cite|improve this answer









$endgroup$



Define $p=lambda x$ and $q=(1-lambda)y$, therefore we need to show that $$||p+q||^2le (||p||+||q||)^2$$which reduces to $$pcdot qle ||p||cdot ||q||$$which is the same famous Cauchy-Schwartz inequality. Therefore $||.||^2$ is convex.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Nov 30 '18 at 11:25









Mostafa AyazMostafa Ayaz

14.7k3938




14.7k3938












  • $begingroup$
    That's fine too!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31










  • $begingroup$
    Thank you. Good luck!
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:33










  • $begingroup$
    This only works if the norm comes from a scalar product, however.
    $endgroup$
    – Giuseppe Negro
    Nov 30 '18 at 11:50


















  • $begingroup$
    That's fine too!
    $endgroup$
    – Omojola Micheal
    Nov 30 '18 at 11:31










  • $begingroup$
    Thank you. Good luck!
    $endgroup$
    – Mostafa Ayaz
    Nov 30 '18 at 11:33










  • $begingroup$
    This only works if the norm comes from a scalar product, however.
    $endgroup$
    – Giuseppe Negro
    Nov 30 '18 at 11:50
















$begingroup$
That's fine too!
$endgroup$
– Omojola Micheal
Nov 30 '18 at 11:31




$begingroup$
That's fine too!
$endgroup$
– Omojola Micheal
Nov 30 '18 at 11:31












$begingroup$
Thank you. Good luck!
$endgroup$
– Mostafa Ayaz
Nov 30 '18 at 11:33




$begingroup$
Thank you. Good luck!
$endgroup$
– Mostafa Ayaz
Nov 30 '18 at 11:33












$begingroup$
This only works if the norm comes from a scalar product, however.
$endgroup$
– Giuseppe Negro
Nov 30 '18 at 11:50




$begingroup$
This only works if the norm comes from a scalar product, however.
$endgroup$
– Giuseppe Negro
Nov 30 '18 at 11:50


















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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3019935%2fprove-that-vert-cdot-vert2x-to-bbbr-where-x-is-a-vector-space-is-co%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

Bundesstraße 106

Image
Dieser Artikel beschreibt die Bundesstraße 106 in Deutschland. Zur gleichnamigen Straße in Österreich siehe Mölltal Straße. Vorlage:Infobox hochrangige Straße/Wartung/DE-B Bundesstraße 106 in Deutschland Karte Basisdaten Betreiber: Deutschland  Bundesrepublik Deutschland Straßenbeginn: Wismar ( 53° 54′  N , 11° 24′  O 53.900536 11.396875 ) Straßenende: Schwerin ( 53° 36′  N , 11° 24′  O 53.592785 11.404892 ) Gesamtlänge: 40,7 km Bundesland: Mecklenburg-Vorpommern Ausbauzustand: zweistreifig [1] Bundesstraße 106 in Schwerin (Umgehungsstraße) Straßenverlauf Land Mecklenburg-Vorpommern Landkreis Nordwestmecklenburg Ortsumgehung Wismar Wallensteingraben über   Dorf Mecklenburg Wallensteingraben Groß Stieten Bad Kleinen OT Niendorf Bahnstrecke Lübeck–Bad Kleinen Zickhusen Lübstorf Klein Trebbow OT Kirch Stück ...
Read more

Verónica Boquete

Image
Verónica Boquete im Trikot der Nationalmannschaft (2013) Personalia Name Verónica Boquete Giadans Geburtstag 9. April 1987 Geburtsort Santiago de Compostela, Spanien Größe 159 cm Position Mittelfeld / Sturm Juniorinnen Jahre Station 2001–2005 Xuventú Aguiño Frauen Jahre Station Spiele (Tore) 1 2005–2008 Prainsa Zaragoza 83 (40) 2008–2010 Espanyol Barcelona 53 (30) 2010 Buffalo Flash 9 0 (9) 2010 Chicago Red Stars 3 0 (1) 2010–2011 Espanyol Barcelona 26 (41) 2011 Philadelphia Independence 13 0 (5) 2011 Energija Woronesch 5 0 (1) 2012–2014 Tyresö FF 39 (14) 2014 Portland Thorns FC 15 0 (4) 2014–2015 1. FFC Frankfurt 23 0 (7) 2015–2016 FC Bayern München 9 0 (0) 2016–2018 Paris Saint-Germain 31 0 (8) 2018 Peking BG Phoenix 2019– Utah Royals FC 0 0 (0) Nationalmannschaft Jahre Auswahl ...
Read more

Ida-Boy-Ed-Garten

Image
Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
Read more