Distance from Vector to the Linear Span

Question

Let $V$ be the space of real polynomials of degree $\leq n$.

a) Check the setting $(f(x),\,g(x))=\int_{0}^{1}f(x)g(x)\,dx$ turns $V$ to a Euclidean space.

b) If $n=1$, find the distance from $f(x)=1$ to the linear span $U=\langle x\rangle$.

Answer 1

Hint: Due to Pythagoras the square of the distance from $v$ to $\mathrm{span}(u)$ is $\|v\|^2-\langle\frac{u}{\|u\|},v\rangle^2$.