Scalar multiplication is defined generally as a function $\displaystyle S:E \times E \longrightarrow K$,where $\displaystyle K\in(R,C)$, $\displaystyle E$ - linear space, for which 5 axioms are true:

$\displaystyle 1. S(x+y,z)=S(x,z)+S(y,z)\\2. S(\lambda x,y)=\lambda S(x,y)\\3. S(x,y)=\bar {S(x,y)}\\4. S(x,x)\geq 0\\5. S(x,x)=0 \Rightarrow x=\bar{0}$

It is neccessary for any function that defines scalar multiplication, that all of these axioms are true.

I've noticed that some sources offer just 4 axioms - the first and the second is joined into one. Does it mean that the first and the second axiom is equivalent?

If not, then there must be a function for which the first axiom is false, but the remaining ones is true. But I cannot think of such an example.