Norm of a function?

I'm confused about norms. I have

1) $\displaystyle A=1-ic\sin\theta$

2)$\displaystyle \parallel A \parallel = 1+c^2\sin^2\theta$

How does one get from 1) to 2)? What kind of norm would this be? (i is imaginary number)

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Originally Posted by Jason Bourne

I'm confused about norms. I have

1) $\displaystyle A=1-ic\sin\theta$

2)$\displaystyle \parallel A \parallel = \color{blue}\sqrt{1+c^2\sin^2\theta}$

How does one get from 1) to 2)? What kind of norm would this be? (i is imaginary number)

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First, note the correction above.

It is the case that for any complex number $\displaystyle \|a+b\mathbf{i}\|=\sqrt{a^2+b^2}$

Also note than many textbooks use $\displaystyle |A|$ instead of $\displaystyle \|A\|$.