Prove that $T$ is a linear operator
$begingroup$
Let $(v_n)_{ngeqslant1}$ ∈ $l^2$ be a fixed sequence of real numbers.
Define a mapping T on $X=l^infty$ using the formula
$T(a_1,a_2,...) = (v_1a_1,v_2a_2,...),$ $x=(a_1,a_2,...)∈ l^infty$.
I have already proved that $Tx ∈ l^2$ for every $x∈l^infty$ for the first part of this question, the next part asks me to prove that
$$T:(l^infty, ||⋅||_infty)rightarrow (l^2,||⋅||_2)$$
is a linear operator.
real-analysis functional-analysis
edited Dec 10 '18 at 3:50
user1101010
7171730
asked Dec 9 '18 at 21:20
callistacallista
32
$endgroup$
add a comment |
$begingroup$
Let $(v_n)_{ngeqslant1}$ ∈ $l^2$ be a fixed sequence of real numbers.
Define a mapping T on $X=l^infty$ using the formula
$T(a_1,a_2,...) = (v_1a_1,v_2a_2,...),$ $x=(a_1,a_2,...)∈ l^infty$.
I have already proved that $Tx ∈ l^2$ for every $x∈l^infty$ for the first part of this question, the next part asks me to prove that
$$T:(l^infty, ||⋅||_infty)rightarrow (l^2,||⋅||_2)$$
is a linear operator.
real-analysis functional-analysis
edited Dec 10 '18 at 3:50
user1101010
7171730
asked Dec 9 '18 at 21:20
callistacallista
32
$endgroup$
$begingroup$
Is your question on the next part? What is your question? Also if that is your question, what have you tried?
$endgroup$
– jgon
Dec 9 '18 at 21:29
$begingroup$
Sorry if it wasn't clear. My question is the next part, verifying that T is a linear operator, the notes I have aren't clear but I've tried to do this using the definition but it's not really working out.
$endgroup$
– callista
Dec 9 '18 at 21:46
add a comment |
$begingroup$
Let $(v_n)_{ngeqslant1}$ ∈ $l^2$ be a fixed sequence of real numbers.
Define a mapping T on $X=l^infty$ using the formula
$T(a_1,a_2,...) = (v_1a_1,v_2a_2,...),$ $x=(a_1,a_2,...)∈ l^infty$.
I have already proved that $Tx ∈ l^2$ for every $x∈l^infty$ for the first part of this question, the next part asks me to prove that
$$T:(l^infty, ||⋅||_infty)rightarrow (l^2,||⋅||_2)$$
is a linear operator.
real-analysis functional-analysis
edited Dec 10 '18 at 3:50
user1101010
7171730
asked Dec 9 '18 at 21:20
callistacallista
32
$endgroup$
Let $(v_n)_{ngeqslant1}$ ∈ $l^2$ be a fixed sequence of real numbers.
Define a mapping T on $X=l^infty$ using the formula
$T(a_1,a_2,...) = (v_1a_1,v_2a_2,...),$ $x=(a_1,a_2,...)∈ l^infty$.
I have already proved that $Tx ∈ l^2$ for every $x∈l^infty$ for the first part of this question, the next part asks me to prove that
$$T:(l^infty, ||⋅||_infty)rightarrow (l^2,||⋅||_2)$$
is a linear operator.
real-analysis functional-analysis
real-analysis functional-analysis
edited Dec 10 '18 at 3:50
user1101010
7171730
asked Dec 9 '18 at 21:20
callistacallista
32
edited Dec 10 '18 at 3:50
user1101010
7171730
asked Dec 9 '18 at 21:20
callistacallista
32
edited Dec 10 '18 at 3:50
user1101010
7171730
edited Dec 10 '18 at 3:50
user1101010
7171730
edited Dec 10 '18 at 3:50
user1101010
7171730
7171730
asked Dec 9 '18 at 21:20
callistacallista
32
asked Dec 9 '18 at 21:20
callistacallista
32
asked Dec 9 '18 at 21:20
callistacallista
32
32
$begingroup$
Is your question on the next part? What is your question? Also if that is your question, what have you tried?
$endgroup$
– jgon
Dec 9 '18 at 21:29
$begingroup$
Sorry if it wasn't clear. My question is the next part, verifying that T is a linear operator, the notes I have aren't clear but I've tried to do this using the definition but it's not really working out.
$endgroup$
– callista
Dec 9 '18 at 21:46
add a comment |
$begingroup$
Is your question on the next part? What is your question? Also if that is your question, what have you tried?
$endgroup$
– jgon
Dec 9 '18 at 21:29
$begingroup$
Sorry if it wasn't clear. My question is the next part, verifying that T is a linear operator, the notes I have aren't clear but I've tried to do this using the definition but it's not really working out.
$endgroup$
– callista
Dec 9 '18 at 21:46
$begingroup$
Is your question on the next part? What is your question? Also if that is your question, what have you tried?
$endgroup$
– jgon
Dec 9 '18 at 21:29
$begingroup$
Is your question on the next part? What is your question? Also if that is your question, what have you tried?
$endgroup$
– jgon
Dec 9 '18 at 21:29
$begingroup$
Sorry if it wasn't clear. My question is the next part, verifying that T is a linear operator, the notes I have aren't clear but I've tried to do this using the definition but it's not really working out.
$endgroup$
– callista
Dec 9 '18 at 21:46
$begingroup$
Sorry if it wasn't clear. My question is the next part, verifying that T is a linear operator, the notes I have aren't clear but I've tried to do this using the definition but it's not really working out.
$endgroup$
– callista
Dec 9 '18 at 21:46
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Hint:
The operator $T$ is linear if the following hold for $s,tin l^infty$, $cinmathbb{R}$:
1) $quad$ $T(s+t) = T(s) + T(t)$
2) $quad$ $T(ccdot s) = ccdot T(s)$
So
$T(s+t) = (v_1(s_1+t_1),v_2(s_1+t_1),...)$ by definition of the operator, can you gontinue from here?
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
$endgroup$
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
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
$begingroup$
Hint:
The operator $T$ is linear if the following hold for $s,tin l^infty$, $cinmathbb{R}$:
1) $quad$ $T(s+t) = T(s) + T(t)$
2) $quad$ $T(ccdot s) = ccdot T(s)$
So
$T(s+t) = (v_1(s_1+t_1),v_2(s_1+t_1),...)$ by definition of the operator, can you gontinue from here?
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
$endgroup$
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
add a comment |
$begingroup$
Hint:
The operator $T$ is linear if the following hold for $s,tin l^infty$, $cinmathbb{R}$:
1) $quad$ $T(s+t) = T(s) + T(t)$
2) $quad$ $T(ccdot s) = ccdot T(s)$
So
$T(s+t) = (v_1(s_1+t_1),v_2(s_1+t_1),...)$ by definition of the operator, can you gontinue from here?
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
$endgroup$
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
add a comment |
$begingroup$
Hint:
The operator $T$ is linear if the following hold for $s,tin l^infty$, $cinmathbb{R}$:
1) $quad$ $T(s+t) = T(s) + T(t)$
2) $quad$ $T(ccdot s) = ccdot T(s)$
So
$T(s+t) = (v_1(s_1+t_1),v_2(s_1+t_1),...)$ by definition of the operator, can you gontinue from here?
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
$endgroup$
Hint:
The operator $T$ is linear if the following hold for $s,tin l^infty$, $cinmathbb{R}$:
1) $quad$ $T(s+t) = T(s) + T(t)$
2) $quad$ $T(ccdot s) = ccdot T(s)$
So
$T(s+t) = (v_1(s_1+t_1),v_2(s_1+t_1),...)$ by definition of the operator, can you gontinue from here?
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
answered Dec 9 '18 at 21:36
Bo5manBo5man
517
517
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
add a comment |
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Oh right, so do I only really need to show that the conditions of linearity are met?
$endgroup$
– callista
Dec 9 '18 at 21:50
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
$begingroup$
Yes, that is right.
$endgroup$
– Bo5man
Dec 10 '18 at 8:48
add a comment |
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$begingroup$
Is your question on the next part? What is your question? Also if that is your question, what have you tried?
$endgroup$
– jgon
Dec 9 '18 at 21:29
$begingroup$
Sorry if it wasn't clear. My question is the next part, verifying that T is a linear operator, the notes I have aren't clear but I've tried to do this using the definition but it's not really working out.
$endgroup$
– callista
Dec 9 '18 at 21:46