One of the properties for inner product says:
$$\left \langle \lambda u,v \right \rangle = \lambda\left \langle u,v \right \rangle$$ for all scalar lambda. $$\left \langle u_{1}+u_{2},v \right \rangle = \left \langle u_{1},v \right \rangle + \left \langle u_{2},v \right \rangle$$
but why? Is there a proof for it? In particular, why isn't the scalar lambda multiplied to both u and v?
On
That's the definition of an inner product. There is no proof for definitions. Asking why is the inner product linear in the first argument is lika asking why triangles have three sides.
The question "why do we take such a definition" is different. The answer to that is at least in some way subjective:
Because it is useful to us.
Many vector spaces we encounter have a "natural" inner product associated with them. $\mathbb R^n$ has a natural inner product which is very useful, since, for example, $\langle x,x\rangle$ is actually the square of the length of $x$.
The second equation is part of the definition of being linear in the first argument. So there is no proof, an inner product is defined to be linear in the first argument.
See Inner product space -> Definition
But if you want to prove something there is still hope: If you get a function and your task is to prove that it is an inner product, then you really have to show that it is linear in the first argument.
Example: Show that $\langle x,y\rangle := x y$ defines an inner product on the $\mathbb R$-vector space $\mathbb R$.