Briefly explain the Gram-Schmidt orthogonalisation process in a real finite-dimensional inner product space $V$.

For a subspace $U$ of $V$, define $U^{\perp}$, and show that $V=U \oplus U^{\perp}$.

For which positive integers $n$ does

$$(f, g)=f(1) g(1)+f(2) g(2)+f(3) g(3)$$

define an inner product on the space of all real polynomials of degree at most $n$ ?