Show that this defines a norm on $mathbbR^n$. State and prove the Holder and Minkowski Inequalities for this norm. The Next CEO of Stack OverflowNorm of a particular step functionShow $frac1^p-frac1in L^2(mathbbR)$ for fixed real yIs it possible to show that the addition of two Cauchy sequences in $mathbb R^n$ is also Cauchy for any metric?State and prove conditions for $|x|_a=sum_j=1^n a_jlvert x_jrvert$ to be a norm on $mathbb R^n$Show that $f_nto f$ in the norm $L^1(mathbbR)$ for $fin L^1(mathbbR)$.The Minkowski inequality for fractional order?How to show that this norm is a metric?Show that linear operator is bounded and calculate its normHow to show that $inf f(I)>0$ for an interval $I$ and a function $f$ with the following property?Show that $hatf$ is integrable over $mathbbR$ and $int_E f$=$int_mathbbR hatf$.
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Show that this defines a norm on $mathbbR^n$. State and prove the Holder and Minkowski Inequalities for this norm.
The Next CEO of Stack OverflowNorm of a particular step functionShow $frac1x-y-frac1in L^2(mathbbR)$ for fixed real yIs it possible to show that the addition of two Cauchy sequences in $mathbb R^n$ is also Cauchy for any metric?State and prove conditions for $|x|_a=sum_j=1^n a_jlvert x_jrvert$ to be a norm on $mathbb R^n$Show that $f_nto f$ in the norm $L^1(mathbbR)$ for $fin L^1(mathbbR)$.The Minkowski inequality for fractional order?How to show that this norm is a metric?Show that linear operator is bounded and calculate its normHow to show that $inf f(I)>0$ for an interval $I$ and a function $f$ with the following property?Show that $hatf$ is integrable over $mathbbR$ and $int_E f$=$int_mathbbR hatf$.
$begingroup$
For a point x = ($x_1$, $x_2$, ... ,$x_n$) in $mathbbR^n$, define $T_x$ to be the step function on the interval [1, n + 1) that takes the value $x_k$ on the interval [k, k + 1), for 1 $leq$ k $leq$ n. For p $geq$ 1, define $| x|_p$ = $| T_x|_p$, the norm of the function $T_x$ in $L^p$[1, n + 1). Show that this defines a norm on $mathbbR^n$. State and prove the Holder and Minkowski Inequalities for this norm.
I know this question has been posted before, however, no straightforward answers were provided. I know that in order to show something is a norm, we need to show the four conditions. However, I don't know how to do that in relation to this problem. Any concrete steps/answers are appreciated.
real-analysis analysis
asked Mar 27 at 18:31
Sawyer Sawyer
92
$endgroup$
add a comment |
$begingroup$
For a point x = ($x_1$, $x_2$, ... ,$x_n$) in $mathbbR^n$, define $T_x$ to be the step function on the interval [1, n + 1) that takes the value $x_k$ on the interval [k, k + 1), for 1 $leq$ k $leq$ n. For p $geq$ 1, define $| x|_p$ = $| T_x|_p$, the norm of the function $T_x$ in $L^p$[1, n + 1). Show that this defines a norm on $mathbbR^n$. State and prove the Holder and Minkowski Inequalities for this norm.
I know this question has been posted before, however, no straightforward answers were provided. I know that in order to show something is a norm, we need to show the four conditions. However, I don't know how to do that in relation to this problem. Any concrete steps/answers are appreciated.
real-analysis analysis
asked Mar 27 at 18:31
Sawyer Sawyer
92
$endgroup$
$begingroup$
What are your problems? What have you done so far?
$endgroup$
– amsmath
Mar 27 at 18:40
$begingroup$
Honestly, the step function confuses me. It's difficult to see what the norm is when a step function is involved.
$endgroup$
– Sawyer
Mar 27 at 19:20
$begingroup$
This is pretty simple. The norm is just the usual $p$-norm for vectors: $|x|_p = left(sum_k=1^n|x_k|^pright)^1/p$.
$endgroup$
– amsmath
Mar 27 at 19:28
add a comment |
$begingroup$
For a point x = ($x_1$, $x_2$, ... ,$x_n$) in $mathbbR^n$, define $T_x$ to be the step function on the interval [1, n + 1) that takes the value $x_k$ on the interval [k, k + 1), for 1 $leq$ k $leq$ n. For p $geq$ 1, define $| x|_p$ = $| T_x|_p$, the norm of the function $T_x$ in $L^p$[1, n + 1). Show that this defines a norm on $mathbbR^n$. State and prove the Holder and Minkowski Inequalities for this norm.
I know this question has been posted before, however, no straightforward answers were provided. I know that in order to show something is a norm, we need to show the four conditions. However, I don't know how to do that in relation to this problem. Any concrete steps/answers are appreciated.
real-analysis analysis
asked Mar 27 at 18:31
Sawyer Sawyer
92
$endgroup$
For a point x = ($x_1$, $x_2$, ... ,$x_n$) in $mathbbR^n$, define $T_x$ to be the step function on the interval [1, n + 1) that takes the value $x_k$ on the interval [k, k + 1), for 1 $leq$ k $leq$ n. For p $geq$ 1, define $| x|_p$ = $| T_x|_p$, the norm of the function $T_x$ in $L^p$[1, n + 1). Show that this defines a norm on $mathbbR^n$. State and prove the Holder and Minkowski Inequalities for this norm.
I know this question has been posted before, however, no straightforward answers were provided. I know that in order to show something is a norm, we need to show the four conditions. However, I don't know how to do that in relation to this problem. Any concrete steps/answers are appreciated.
real-analysis analysis
real-analysis analysis
asked Mar 27 at 18:31
Sawyer Sawyer
92
asked Mar 27 at 18:31
Sawyer Sawyer
92
asked Mar 27 at 18:31
Sawyer Sawyer
92
asked Mar 27 at 18:31
Sawyer Sawyer
92
asked Mar 27 at 18:31
Sawyer Sawyer
92
92
$begingroup$
What are your problems? What have you done so far?
$endgroup$
– amsmath
Mar 27 at 18:40
$begingroup$
Honestly, the step function confuses me. It's difficult to see what the norm is when a step function is involved.
$endgroup$
– Sawyer
Mar 27 at 19:20
$begingroup$
This is pretty simple. The norm is just the usual $p$-norm for vectors: $|x|_p = left(sum_k=1^n|x_k|^pright)^1/p$.
$endgroup$
– amsmath
Mar 27 at 19:28
add a comment |
$begingroup$
What are your problems? What have you done so far?
$endgroup$
– amsmath
Mar 27 at 18:40
$begingroup$
Honestly, the step function confuses me. It's difficult to see what the norm is when a step function is involved.
$endgroup$
– Sawyer
Mar 27 at 19:20
$begingroup$
This is pretty simple. The norm is just the usual $p$-norm for vectors: $|x|_p = left(sum_k=1^n|x_k|^pright)^1/p$.
$endgroup$
– amsmath
Mar 27 at 19:28
$begingroup$
What are your problems? What have you done so far?
$endgroup$
– amsmath
Mar 27 at 18:40
$begingroup$
What are your problems? What have you done so far?
$endgroup$
– amsmath
Mar 27 at 18:40
$begingroup$
Honestly, the step function confuses me. It's difficult to see what the norm is when a step function is involved.
$endgroup$
– Sawyer
Mar 27 at 19:20
$begingroup$
Honestly, the step function confuses me. It's difficult to see what the norm is when a step function is involved.
$endgroup$
– Sawyer
Mar 27 at 19:20
$begingroup$
This is pretty simple. The norm is just the usual $p$-norm for vectors: $|x|_p = left(sum_k=1^n|x_k|^pright)^1/p$.
$endgroup$
– amsmath
Mar 27 at 19:28
$begingroup$
This is pretty simple. The norm is just the usual $p$-norm for vectors: $|x|_p = left(sum_k=1^n|x_k|^pright)^1/p$.
$endgroup$
– amsmath
Mar 27 at 19:28
add a comment |
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$begingroup$
What are your problems? What have you done so far?
$endgroup$
– amsmath
Mar 27 at 18:40
$begingroup$
Honestly, the step function confuses me. It's difficult to see what the norm is when a step function is involved.
$endgroup$
– Sawyer
Mar 27 at 19:20
$begingroup$
This is pretty simple. The norm is just the usual $p$-norm for vectors: $|x|_p = left(sum_k=1^n|x_k|^pright)^1/p$.
$endgroup$
– amsmath
Mar 27 at 19:28