Improved part to this question
Let $V$ be the space of real polynomials in one variable $t$ of degree less than or equal to three. Define $$ \langle p,q\rangle = p(1)q(1)+p'(1)q'(1)+p''(1)q''(1)+p'''(1)q'''(1). $$
(i) Prove that $\langle\cdot,\cdot\rangle$ defines an inner product.
Here is what I had:
Consider $P_t$, the space of polynomial of degree polynomial at most $t$ ( with coefficients from R)Let $x_0<x_1<x_2<…<x_3$ be any $t+1$ real numbers. For $p$ and q belong to $P_t$ ,we define $<p,q>$ to be $p(1)q(1)+p(1)'q(1)'+p(1)''q(1)''+p(1)'''q(1)'''$ is an inner product on $P_t$. Checking that $<p,P> = 0 $->$ p = 0$ then $1$ is the root of $p$, and as a polynomial of degree at most $t$ has at most $t$ roots unless it's the zero polynomial, p must be the zero polynomial. Different choice of $1$ gives rise to different inner products.
Could we just do this $f(a)=0$ and $f'(a)=0$ then $f(x)$ is divisible by $(x-a)^2$ ?
Can someone please help me with this proof for part (i). It is frustrating me.