Polynomial Interpolation question

Question

I do not understand the part that I have underlined in green. I thought that if you fit a polynomial $P_n$to given data for say $n+1$ distinct points you got an approximation of $f(x)$, where there is an error $f-P_n$.

Answer 1

In general fitting a polynomial to points is a pretty bad way to approximate a function unless the function is a polynomial. Usually the Taylor polynomials are much better. The only way the underlined part you mentioned makes sense is if $f$ is a polynomial, otherwise the norm $||f - p||$ will not be zero.