For $(x_1,x_2), (y_1,y_2) \in \mathbb{R}^2$, does $⟨(x_1,x_2),(y_1,y_2)⟩={x_1}^2{y_1}^2 + {x_2}^2{y_2}^2$ define an inner product?

Question

For $(x_1,x_2), (y_1,y_2) \in \mathbb{R}^2,$ let $⟨(x_1,x_2),(y_1,y_2)⟩={x_1}^2{y_1}^2 + {x_2}^2{y_2}^2$

Does this define an inner product? I know an inner product has to have linearity, be symmetric and positive definite i.e. $⟨ u, u⟩ > 0$ if and only if u = 0. It is clearly symmetric and positive definite, however I am struggling with showing if it is linear or not. Any guidance on this would be appreciated.

Answer 1

Since $\langle(2,0),(1,0)\rangle=4\ne2\langle(1,0),(1,0)\rangle$, no, it is not linear in the first variable, and therefore $\langle\cdot,\cdot\rangle$ is not an inner product.