Q:

What is the norm of a complex vector?

I have two arrays $a$ and $b$ containing complex values. Now I one of my target operations is the following:
$$||a-b||$$
The result should be a single real number. Now I am a bit confused how to apply the norm here: Should I rather calculate first the norm of each value in the array, and then apply the formula above:
$$a[i] = \sqrt(a[i].real^2+a[i].imag^2)$$
$$||a[i]-b[i]||$$
or should I first subtract the values, and then apply the formula:
$$tmp = a[i]-b[i]$$
$$||tmp|| = \left(\sum \sqrt(tmp.real^2 + tmp.imag^2)^n\right)^{\frac{1}{n}}$$?
What is intended by the operation shown above?

A:

For any complex vector $$\|c\|=\sqrt{\langle c,c\rangle}=\sqrt{\sum_{i=1}^nc_i\overline{c}_i}.$$

