MHB Are the Weak and Strong Norms Correct for These Function Distances?

[evinda]

Hello! (Wave)

In $C^1[0,1]$ calculate the distances between the functions $y_1(x)=0$ and $y_2(x)=\frac{1}{100} \sin(1000x)$ in respect to the weak and strong norm.That's what I have tried:Weak norm:

$||y_1-y_2||_w=\max_{0 \leq x \leq 1} |y_1(x)-y_2(x)|+ \max_{0 \leq x \leq 1} |y_1'(x)-y_2'(x) | \\=\max_{0 \leq x \leq 1} \left |0-\frac{1}{100} \sin(1000x) \right|+ \max_{0 \leq x \leq 1} \left |\frac{1000}{100} \cos(1000x) \right|= \frac{1}{100}+10=\frac{1001}{100}$

Strong norm:

$||y_1-y_2||_M= \max_{0 \leq x \leq 1} |y_1(x)-y_2(x)|=\max_{0 \leq x \leq 1} \left |y_1(x)-y_2(x) \right|= \\=\max_{0 \leq x \leq 1} \left |0-\frac{1}{100} \sin(1000x) \right|= \frac{1}{100} \max_{0 \leq x \leq 1} |\sin(1000x)|= \frac{1}{100}$

Could you tell me if that what I have tried is right? (Thinking)

 

[Opalg]

Hello! (Wave)

In $C^1[0,1]$ calculate the distances between the functions $y_1(x)=0$ and $y_2(x)=\frac{1}{100} \sin(1000x)$ in respect to the weak and strong norm.That's what I have tried:Weak norm:

$||y_1-y_2||_w=\max_{0 \leq x \leq 1} |y_1(x)-y_2(x)|+ \max_{0 \leq x \leq 1} |y_1'(x)-y_2'(x) | \\=\max_{0 \leq x \leq 1} \left |0-\frac{1}{100} \sin(1000x) \right|+ \max_{0 \leq x \leq 1} \left |\frac{1000}{100} \cos(1000x) \right|= \frac{1}{100}+10=\frac{1001}{100}$

Strong norm:

$||y_1-y_2||_M= \max_{0 \leq x \leq 1} |y_1(x)-y_2(x)|=\max_{0 \leq x \leq 1} \left |y_1(x)-y_2(x) \right|= \\=\max_{0 \leq x \leq 1} \left |0-\frac{1}{100} \sin(1000x) \right|= \frac{1}{100} \max_{0 \leq x \leq 1} |\sin(1000x)|= \frac{1}{100}$

Could you tell me if that what I have tried is right? (Thinking)

If your definitions of the weak and strong norms are correct then so are those answers. But the definitions look odd to me because the "weak" norm is stronger than the "strong" norm.

 

[evinda]

If your definitions of the weak and strong norms are correct then so are those answers. But the definitions look odd to me because the "weak" norm is stronger than the "strong" norm.

According to my textbook:In the linear space $C^1[a,b]$, the norm that is defined as

$$||y||_w=\max_{a \leq x \leq b} |y(x)|+ \max_{a \leq x \leq b} |y'(x)|$$

is called weak norm.

The maximum norm that we define like that: $||y||_M=\max_{a \leq x \leq b} |y(x)| dx$ is called strong norm.

 

[evinda]

If your definitions of the weak and strong norms are correct then so are those answers. But the definitions look odd to me because the "weak" norm is stronger than the "strong" norm.

So do you think that it is a typo? (Thinking)