Inner product with the given property and vector distance problem

Question

Is there an inner product in the $P_2$ ( polynomials, $deg(p)\leq 2$)inner product space so that $\{1,t,t^2\}$ is an orthonormal base?

Also, how do you find a subspace for a given vector and a distance between the given vector and subspace? (example for the given vector $v=(2,3,0,4)$ find a subpace $S$ so that the distance is 1)

Thank you!!

Answer 1

In $P_2 $ consider the inner product $$\left< at^2 +bt +c , dt^2 +et +f \right> =ad +be +cf. $$