Define the semiscalar product [x,y] by $ $ [x,y]=\inf_{t>0}\frac{1}{2t}[||x+ty||^2-||x||^2].$ $ E be an n.v.s. I donot know how to prove that
1.$ [x,\lambda x+\mu y]=\lambda||x||^2+\mu[x,y]\ \forall x,y\in E,\lambda\in\mathbb{R},\forall \mu >0.$
2.$ [\lambda x,\mu y]=\lambda\mu[x,y],\forall x,y\in E,\forall\lambda,\mu\geq 0$
I probably understand that this semiscalar product is similar to an inner product, but I only have trigonometric inequalities for its norm, how do I prove that this product is linear?