In my classwork I have to provide examples of infinite dimensional vector spaces with metric generated by their norm. I readily provide space of continuous function $C[a, b]$ and Hilbert sequence space $\mathscr l^2$ with their respective norm as the easiest examples, but I am still looking for a few more simple examples.
From this old MSE link here I see that the function space $L^p$ and the polynomials in one variable $\mathbb R[x]$ are also simple examples. I know from other resources that the norm to the function space $L^p$ is
$$\lVert f \rVert_p = \left( \int^a_b \lvert f(x) \rvert^p dx \right)^{\frac{1}{p}}$$ which I think is capable of generating a metric by $ d(f, g) = \lVert f - g \rVert_p$.
And here is my question: Does anyone know of a norm to the above polynomials in one variable $\mathbb R [x]$ that is capable of generating a metric? Thanks for your time and effort.
