To write this sentence about a distribution more rigorously

Question

I have the sentence at the moment

Notice that for all $c \in \mathbf{C}$ such that $W(cx) = |c|^{2} Wx$.

which I do not like.

I mean to say that for all $c \in \mathbf C$ the equation is true.

Probably simply

Notice for all $c \in \mathbf C, W(cx) = |c|^{2} Wx.$

How can you formulate the sentence about the distribution equation better, when it should stay as a notification only?

Before I have defined this:

$W: L^{2}(\mathbf R) \to L^{2}(\mathbf R \times \mathbf R)$

Answer 1

My offer would be

Note that $W(cx) = \lvert c\rvert^2 Wx$ for all $c \in \mathbf{C}$.

Answer 2

How about the following?

Notice that $A=B$ holds for all $C$.