Inner product axioms

Bilinearity. For $u,v,w\in\mathbb{R}^2$ and $\lambda\in\mathbb{R}$:
$$\langle u+\lambda w,v\rangle = 2(u_1+\lambda w_1)v_1+(u_2+\lambda w_2)v_2 = \langle u,v\rangle+\lambda\langle w,v\rangle.$$
Symmetry in the second variable follows similarly.

Symmetry. $\langle u,v\rangle = 2u_1v_1+u_2v_2 = 2v_1u_1+v_2u_2=\langle v,u\rangle$.

Positive definiteness. $\langle u,u\rangle = 2u_1^2+u_2^2\ge0$, with equality iff $u_1=u_2=0$.

Conclusion. $\langle\cdot,\cdot\rangle$ is an inner product on $\mathbb{R}^2$.