Answer
the function is not an inner product.
Work Step by Step
Since $\langle (u_1,u_2), (v_1, v_2 )\rangle= u_1v_1 $ and $\langle (v_1, v_2 ),(u_1,u_2)\rangle = u_2v_2 $, then we get
$$\langle (u_1,u_2), (v_1, v_2 )\rangle\neq \langle (v_1, v_2 ), (u_1,u_2)\rangle.$$
This means that the function is not an inner product.
