Mixing
find adjoint operator of an operator A
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How to find adjoint operator of an operator A
$$A in B(C^1[0,1], C[0,1])$$ $$ (Ax)(t) = x'(t)?$$
In answer : for any functional $f_y$ originated by function $y in BV_0[0,1]:A(f'_y) = g_z$, where functional $g_z$ originated by couple of function $z(t) = y(t)$ and number zero.
Have no idea how to find. Can you help me with this?
functional-analysis
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
$endgroup$
|
show 2 more comments
$begingroup$
How to find adjoint operator of an operator A
$$A in B(C^1[0,1], C[0,1])$$ $$ (Ax)(t) = x'(t)?$$
In answer : for any functional $f_y$ originated by function $y in BV_0[0,1]:A(f'_y) = g_z$, where functional $g_z$ originated by couple of function $z(t) = y(t)$ and number zero.
Have no idea how to find. Can you help me with this?
functional-analysis
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
$endgroup$
$begingroup$
Is $A$ even bounded? What norm are we using?
$endgroup$
– SmileyCraft
Jan 8 at 21:42
$begingroup$
@SmileyCraft for C[0,1] :$parallel x parallel = max | x(t)|$, where $t in [0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:50
$begingroup$
And for $xin C^1[0,1]$?
$endgroup$
– SmileyCraft
Jan 8 at 21:54
$begingroup$
@SmileyCraft $parallel xparallel =max|x'(t)|$, where $tin[0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:56
$begingroup$
Then $A$ is not bounded. Consider $f_n(t)=t^n$. Then $|f_n|=1$, but $|Af_n|=n$, since $f_n'(t)=nt^{n-1}$.
$endgroup$
– SmileyCraft
Jan 8 at 21:58
|
show 2 more comments
$begingroup$
How to find adjoint operator of an operator A
$$A in B(C^1[0,1], C[0,1])$$ $$ (Ax)(t) = x'(t)?$$
In answer : for any functional $f_y$ originated by function $y in BV_0[0,1]:A(f'_y) = g_z$, where functional $g_z$ originated by couple of function $z(t) = y(t)$ and number zero.
Have no idea how to find. Can you help me with this?
functional-analysis
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
$endgroup$
How to find adjoint operator of an operator A
$$A in B(C^1[0,1], C[0,1])$$ $$ (Ax)(t) = x'(t)?$$
In answer : for any functional $f_y$ originated by function $y in BV_0[0,1]:A(f'_y) = g_z$, where functional $g_z$ originated by couple of function $z(t) = y(t)$ and number zero.
Have no idea how to find. Can you help me with this?
functional-analysis
functional-analysis
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
edited Jan 8 at 21:40
Gera Slanova
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
asked Jan 8 at 21:26
Gera SlanovaGera Slanova
453
453
$begingroup$
Is $A$ even bounded? What norm are we using?
$endgroup$
– SmileyCraft
Jan 8 at 21:42
$begingroup$
@SmileyCraft for C[0,1] :$parallel x parallel = max | x(t)|$, where $t in [0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:50
$begingroup$
And for $xin C^1[0,1]$?
$endgroup$
– SmileyCraft
Jan 8 at 21:54
$begingroup$
@SmileyCraft $parallel xparallel =max|x'(t)|$, where $tin[0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:56
$begingroup$
Then $A$ is not bounded. Consider $f_n(t)=t^n$. Then $|f_n|=1$, but $|Af_n|=n$, since $f_n'(t)=nt^{n-1}$.
$endgroup$
– SmileyCraft
Jan 8 at 21:58
|
show 2 more comments
$begingroup$
Is $A$ even bounded? What norm are we using?
$endgroup$
– SmileyCraft
Jan 8 at 21:42
$begingroup$
@SmileyCraft for C[0,1] :$parallel x parallel = max | x(t)|$, where $t in [0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:50
$begingroup$
And for $xin C^1[0,1]$?
$endgroup$
– SmileyCraft
Jan 8 at 21:54
$begingroup$
@SmileyCraft $parallel xparallel =max|x'(t)|$, where $tin[0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:56
$begingroup$
Then $A$ is not bounded. Consider $f_n(t)=t^n$. Then $|f_n|=1$, but $|Af_n|=n$, since $f_n'(t)=nt^{n-1}$.
$endgroup$
– SmileyCraft
Jan 8 at 21:58
$begingroup$
Is $A$ even bounded? What norm are we using?
$endgroup$
– SmileyCraft
Jan 8 at 21:42
$begingroup$
Is $A$ even bounded? What norm are we using?
$endgroup$
– SmileyCraft
Jan 8 at 21:42
$begingroup$
@SmileyCraft for C[0,1] :$parallel x parallel = max | x(t)|$, where $t in [0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:50
$begingroup$
@SmileyCraft for C[0,1] :$parallel x parallel = max | x(t)|$, where $t in [0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:50
$begingroup$
And for $xin C^1[0,1]$?
$endgroup$
– SmileyCraft
Jan 8 at 21:54
$begingroup$
And for $xin C^1[0,1]$?
$endgroup$
– SmileyCraft
Jan 8 at 21:54
$begingroup$
@SmileyCraft $parallel xparallel =max|x'(t)|$, where $tin[0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:56
$begingroup$
@SmileyCraft $parallel xparallel =max|x'(t)|$, where $tin[0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:56
$begingroup$
Then $A$ is not bounded. Consider $f_n(t)=t^n$. Then $|f_n|=1$, but $|Af_n|=n$, since $f_n'(t)=nt^{n-1}$.
$endgroup$
– SmileyCraft
Jan 8 at 21:58
$begingroup$
Then $A$ is not bounded. Consider $f_n(t)=t^n$. Then $|f_n|=1$, but $|Af_n|=n$, since $f_n'(t)=nt^{n-1}$.
$endgroup$
– SmileyCraft
Jan 8 at 21:58
|
show 2 more comments
1 Answer
1
active
oldest
votes
$begingroup$
Take $muin C[0,1]^*$. By definition, $A^*muin C^1[0,1]^*$ is given by
$$
(A^*mu)(f)=mu(Af)=mu(f')=int_{[0,1]},f',dmu.
$$
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Take $muin C[0,1]^*$. By definition, $A^*muin C^1[0,1]^*$ is given by
$$
(A^*mu)(f)=mu(Af)=mu(f')=int_{[0,1]},f',dmu.
$$
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
add a comment |
$begingroup$
Take $muin C[0,1]^*$. By definition, $A^*muin C^1[0,1]^*$ is given by
$$
(A^*mu)(f)=mu(Af)=mu(f')=int_{[0,1]},f',dmu.
$$
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
add a comment |
$begingroup$
Take $muin C[0,1]^*$. By definition, $A^*muin C^1[0,1]^*$ is given by
$$
(A^*mu)(f)=mu(Af)=mu(f')=int_{[0,1]},f',dmu.
$$
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
$endgroup$
Take $muin C[0,1]^*$. By definition, $A^*muin C^1[0,1]^*$ is given by
$$
(A^*mu)(f)=mu(Af)=mu(f')=int_{[0,1]},f',dmu.
$$
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
answered Jan 8 at 21:32
Martin ArgeramiMartin Argerami
126k1182181
126k1182181
add a comment |
add a comment |
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$begingroup$
Is $A$ even bounded? What norm are we using?
$endgroup$
– SmileyCraft
Jan 8 at 21:42
$begingroup$
@SmileyCraft for C[0,1] :$parallel x parallel = max | x(t)|$, where $t in [0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:50
$begingroup$
And for $xin C^1[0,1]$?
$endgroup$
– SmileyCraft
Jan 8 at 21:54
$begingroup$
@SmileyCraft $parallel xparallel =max|x'(t)|$, where $tin[0,1]$
$endgroup$
– Gera Slanova
Jan 8 at 21:56
$begingroup$
Then $A$ is not bounded. Consider $f_n(t)=t^n$. Then $|f_n|=1$, but $|Af_n|=n$, since $f_n'(t)=nt^{n-1}$.
$endgroup$
– SmileyCraft
Jan 8 at 21:58