Proving that $Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert$ defines a norm on $B(X,Y). $
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Let $B(X,Y)$ be the family of all bounded maps from $X$ to $Y,$ normed linear maps. Then, $Vert cdot Vert,$ defined by $Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert,$ for arbitrary $Tin B(X,Y), $ is a norm on $B(X,Y). $
we have for arbitrary $Tin B(X,Y), $
begin{align} Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert= suplimits_{Vert x Vert = 1}Vert T x Vert=suplimits_{xneq 0} frac{Vert T x Vert}{Vert x Vert}.end{align}
MY TRIAL
1.
begin{align} Vert T Vert =0&iff suplimits_{Vert x Vertleq 1}Vert T x Vert =0iff Vert T x Vert =0,;;forall ,xin X, ,Tin B(X,Y)\& iff T x=0,;;forall ,xin X, ,Tin B(X,Y) \& iff T =0,;;forall , ,Tin B(X,Y)end{align}
2.
begin{align} Vert kT Vert =& suplimits_{Vert x Vertleq 1}Vert k T x Vert \=& |k|suplimits_{Vert x Vertleq 1}Vert T x Vert \=& |k|Vert T Vert,;;forall ,,kin K, ,Tin B(X,Y)end{align}
3.
begin{align} Vert T+S Vert =& suplimits_{Vert x Vertleq 1}Vert T x + S x Vert \leq & suplimits_{Vert x Vertleq 1}left(Vert T x Vert + Vert S x Vert right) \=&suplimits_{Vert x Vertleq 1}Vert T x Vert +suplimits_{Vert x Vertleq 1} Vert S x Vert \=&Vert T Vert+Vert S Vert,;;forall ,T,Sin B(X,Y)end{align}
Kindly help check if this is correct. If not, corrections and alternative proofs will be highly welcome.
functional-analysis norm normed-spaces
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$begingroup$
Let $B(X,Y)$ be the family of all bounded maps from $X$ to $Y,$ normed linear maps. Then, $Vert cdot Vert,$ defined by $Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert,$ for arbitrary $Tin B(X,Y), $ is a norm on $B(X,Y). $
we have for arbitrary $Tin B(X,Y), $
begin{align} Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert= suplimits_{Vert x Vert = 1}Vert T x Vert=suplimits_{xneq 0} frac{Vert T x Vert}{Vert x Vert}.end{align}
MY TRIAL
1.
begin{align} Vert T Vert =0&iff suplimits_{Vert x Vertleq 1}Vert T x Vert =0iff Vert T x Vert =0,;;forall ,xin X, ,Tin B(X,Y)\& iff T x=0,;;forall ,xin X, ,Tin B(X,Y) \& iff T =0,;;forall , ,Tin B(X,Y)end{align}
2.
begin{align} Vert kT Vert =& suplimits_{Vert x Vertleq 1}Vert k T x Vert \=& |k|suplimits_{Vert x Vertleq 1}Vert T x Vert \=& |k|Vert T Vert,;;forall ,,kin K, ,Tin B(X,Y)end{align}
3.
begin{align} Vert T+S Vert =& suplimits_{Vert x Vertleq 1}Vert T x + S x Vert \leq & suplimits_{Vert x Vertleq 1}left(Vert T x Vert + Vert S x Vert right) \=&suplimits_{Vert x Vertleq 1}Vert T x Vert +suplimits_{Vert x Vertleq 1} Vert S x Vert \=&Vert T Vert+Vert S Vert,;;forall ,T,Sin B(X,Y)end{align}
Kindly help check if this is correct. If not, corrections and alternative proofs will be highly welcome.
functional-analysis norm normed-spaces
$endgroup$
add a comment |
$begingroup$
Let $B(X,Y)$ be the family of all bounded maps from $X$ to $Y,$ normed linear maps. Then, $Vert cdot Vert,$ defined by $Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert,$ for arbitrary $Tin B(X,Y), $ is a norm on $B(X,Y). $
we have for arbitrary $Tin B(X,Y), $
begin{align} Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert= suplimits_{Vert x Vert = 1}Vert T x Vert=suplimits_{xneq 0} frac{Vert T x Vert}{Vert x Vert}.end{align}
MY TRIAL
1.
begin{align} Vert T Vert =0&iff suplimits_{Vert x Vertleq 1}Vert T x Vert =0iff Vert T x Vert =0,;;forall ,xin X, ,Tin B(X,Y)\& iff T x=0,;;forall ,xin X, ,Tin B(X,Y) \& iff T =0,;;forall , ,Tin B(X,Y)end{align}
2.
begin{align} Vert kT Vert =& suplimits_{Vert x Vertleq 1}Vert k T x Vert \=& |k|suplimits_{Vert x Vertleq 1}Vert T x Vert \=& |k|Vert T Vert,;;forall ,,kin K, ,Tin B(X,Y)end{align}
3.
begin{align} Vert T+S Vert =& suplimits_{Vert x Vertleq 1}Vert T x + S x Vert \leq & suplimits_{Vert x Vertleq 1}left(Vert T x Vert + Vert S x Vert right) \=&suplimits_{Vert x Vertleq 1}Vert T x Vert +suplimits_{Vert x Vertleq 1} Vert S x Vert \=&Vert T Vert+Vert S Vert,;;forall ,T,Sin B(X,Y)end{align}
Kindly help check if this is correct. If not, corrections and alternative proofs will be highly welcome.
functional-analysis norm normed-spaces
$endgroup$
Let $B(X,Y)$ be the family of all bounded maps from $X$ to $Y,$ normed linear maps. Then, $Vert cdot Vert,$ defined by $Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert,$ for arbitrary $Tin B(X,Y), $ is a norm on $B(X,Y). $
we have for arbitrary $Tin B(X,Y), $
begin{align} Vert T Vert =suplimits_{Vert x Vertleq 1}Vert T x Vert= suplimits_{Vert x Vert = 1}Vert T x Vert=suplimits_{xneq 0} frac{Vert T x Vert}{Vert x Vert}.end{align}
MY TRIAL
1.
begin{align} Vert T Vert =0&iff suplimits_{Vert x Vertleq 1}Vert T x Vert =0iff Vert T x Vert =0,;;forall ,xin X, ,Tin B(X,Y)\& iff T x=0,;;forall ,xin X, ,Tin B(X,Y) \& iff T =0,;;forall , ,Tin B(X,Y)end{align}
2.
begin{align} Vert kT Vert =& suplimits_{Vert x Vertleq 1}Vert k T x Vert \=& |k|suplimits_{Vert x Vertleq 1}Vert T x Vert \=& |k|Vert T Vert,;;forall ,,kin K, ,Tin B(X,Y)end{align}
3.
begin{align} Vert T+S Vert =& suplimits_{Vert x Vertleq 1}Vert T x + S x Vert \leq & suplimits_{Vert x Vertleq 1}left(Vert T x Vert + Vert S x Vert right) \=&suplimits_{Vert x Vertleq 1}Vert T x Vert +suplimits_{Vert x Vertleq 1} Vert S x Vert \=&Vert T Vert+Vert S Vert,;;forall ,T,Sin B(X,Y)end{align}
Kindly help check if this is correct. If not, corrections and alternative proofs will be highly welcome.
functional-analysis norm normed-spaces
functional-analysis norm normed-spaces
asked Dec 18 '18 at 9:20
Omojola MichealOmojola Micheal
1,873324
1,873324
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Your proof is fine except that you have to fill in some details on why $sup{|Tx|:|x|leq 1}$ implies $Tx=0$ for all $x$. You get $Tx=0$ for $|x|leq 1$ and then you have to argue that if $x neq 0$ then $frac x {|x|}$ has norm $1$ so $Tfrac x {|x|}=0$; finally, linearity of $T$ gives $frac {Tx} {|x|}=0$, so $Tx=0$.
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Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
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– Omojola Micheal
Dec 18 '18 at 12:06
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Isn't it obvious that $T=0$ implies $|T|=0$?
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– Kavi Rama Murthy
Dec 18 '18 at 23:13
add a comment |
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1 Answer
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$begingroup$
Your proof is fine except that you have to fill in some details on why $sup{|Tx|:|x|leq 1}$ implies $Tx=0$ for all $x$. You get $Tx=0$ for $|x|leq 1$ and then you have to argue that if $x neq 0$ then $frac x {|x|}$ has norm $1$ so $Tfrac x {|x|}=0$; finally, linearity of $T$ gives $frac {Tx} {|x|}=0$, so $Tx=0$.
$endgroup$
$begingroup$
Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
$endgroup$
– Omojola Micheal
Dec 18 '18 at 12:06
$begingroup$
Isn't it obvious that $T=0$ implies $|T|=0$?
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 23:13
add a comment |
$begingroup$
Your proof is fine except that you have to fill in some details on why $sup{|Tx|:|x|leq 1}$ implies $Tx=0$ for all $x$. You get $Tx=0$ for $|x|leq 1$ and then you have to argue that if $x neq 0$ then $frac x {|x|}$ has norm $1$ so $Tfrac x {|x|}=0$; finally, linearity of $T$ gives $frac {Tx} {|x|}=0$, so $Tx=0$.
$endgroup$
$begingroup$
Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
$endgroup$
– Omojola Micheal
Dec 18 '18 at 12:06
$begingroup$
Isn't it obvious that $T=0$ implies $|T|=0$?
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 23:13
add a comment |
$begingroup$
Your proof is fine except that you have to fill in some details on why $sup{|Tx|:|x|leq 1}$ implies $Tx=0$ for all $x$. You get $Tx=0$ for $|x|leq 1$ and then you have to argue that if $x neq 0$ then $frac x {|x|}$ has norm $1$ so $Tfrac x {|x|}=0$; finally, linearity of $T$ gives $frac {Tx} {|x|}=0$, so $Tx=0$.
$endgroup$
Your proof is fine except that you have to fill in some details on why $sup{|Tx|:|x|leq 1}$ implies $Tx=0$ for all $x$. You get $Tx=0$ for $|x|leq 1$ and then you have to argue that if $x neq 0$ then $frac x {|x|}$ has norm $1$ so $Tfrac x {|x|}=0$; finally, linearity of $T$ gives $frac {Tx} {|x|}=0$, so $Tx=0$.
answered Dec 18 '18 at 9:24
Kavi Rama MurthyKavi Rama Murthy
60k42161
60k42161
$begingroup$
Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
$endgroup$
– Omojola Micheal
Dec 18 '18 at 12:06
$begingroup$
Isn't it obvious that $T=0$ implies $|T|=0$?
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 23:13
add a comment |
$begingroup$
Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
$endgroup$
– Omojola Micheal
Dec 18 '18 at 12:06
$begingroup$
Isn't it obvious that $T=0$ implies $|T|=0$?
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 23:13
$begingroup$
Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
$endgroup$
– Omojola Micheal
Dec 18 '18 at 12:06
$begingroup$
Please, how do I get an "if and only if" statement from this? From what you've stated, I think I can only get implication from this. Can you, please, help?
$endgroup$
– Omojola Micheal
Dec 18 '18 at 12:06
$begingroup$
Isn't it obvious that $T=0$ implies $|T|=0$?
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 23:13
$begingroup$
Isn't it obvious that $T=0$ implies $|T|=0$?
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 23:13
add a comment |
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