I am having trouble finding the distance and angle between two polynomials in the space of continuous functions over the interval $[0,1]$. The inner product of this space is defined as:
$$\langle f,g\rangle = \int_{0}^{1} f(t) g(t) dt$$
The two polynomials in question are $p(t) = 1+t$ and $q(t) = 1+t^2$.
I have tried to use the formula for the distance between two vectors in an inner product space:
$$\|p-q\| = \sqrt{\langle p-q, p-q\rangle }$$
However, when I apply this formula to the given polynomials, I end up with a non-zero answer. I suspect that I am doing something wrong or missing a step.
I would appreciate any guidance on how to correctly calculate the distance and angle between these polynomials.
Thank you in advance.