The (global) $L^2$ norm of a discrete scalar field S is easy. Say $S = [a, b, c]$ then $$\lVert S \rVert = \sqrt{a^2 + b^2 + c^2}$$
But how is the (global) $L^2$ norm defined for a 2D vector field V?
Say $V = [(a_x, b_y), (b_x, b_y), (c_x, c_y)]$. I have seen several defintions go by, e.g.:
Definition 1: As the sum of the local $L^2$ norms: $$\lVert V \rVert = \lVert \mathbf{a} \rVert + \lVert \mathbf{b} \rVert + \lVert \mathbf{c} \rVert = \sqrt{a_x^2 + a_y^2} + \sqrt{b_x^2 + b_y^2} + \sqrt{c_x^2 + c_y^2}.$$ Definition 2: As the Frobenius norm (i.e. concatenate x-components and y-components to one vector and compute $L^2$ norm of that new vector): $$\lVert V \rVert = \lVert [a_x, a_y, b_x, b_y, c_x, c_y] \rVert = \sqrt{a_x^2 + a_y^2 + b_x^2 + b_y^2 + c_x^2 + c_y^2}.$$ Definition 3: As the $L^2$ norm of the local $L^2$ norms. This leads to the same results as the second definition: \begin{align*} \lVert V \rVert = \lVert \lVert V_i \rVert \rVert &= \lVert \sqrt{a_x^2 + a_y^2} + \sqrt{b_x^2 + b_y^2} + \sqrt{c_x^2 + c_y^2} \rVert\\ &= \sqrt{\left(\sqrt{a_x^2 + a_y^2}\right)^2 + \left(\sqrt{b_x^2 + b_y^2}\right)^2 + \left(\sqrt{c_x^2 + c_y^2}\right)^2}\\ &= \sqrt{a_x^2 + a_y^2 + b_x^2 + b_y^2 + c_x^2 + c_y^2}. \end{align*} Definition 4: As the sum of the $L^2$ norm of the x-components and the $L^2$ norm of the y-components: $$\lVert V \rVert = \lVert V_x \rVert + \lVert V_y \rVert = \sqrt{a_x^2 + b_x^2 + c_x^2 \vphantom{a_y^2}} + \sqrt{a_y^2 + b_y^2 + c_y^2}.$$ I saw a few more definitions but I think my point is clear.
Is there some (un)written agreement what is the right or common definition?
My specific problem: I have a discrete field containing 2D error vectors, and I want to compute the global $L^2$ norm of this field.