I know that if $f$ is given and $(x_i,y_i), i=0,\ldots,n$ are points of $f$, then the Lagrange interpolating polynomial is $p_n(x)=\sum_{i=0}^n f(x_i)\ell_i$, and given any other interpolating polynomial for $f$ in $(x_i,y_i), i=0,1,\ldots,n$, we have that the Lagrange interpolating polynomial is equivalent to this, thus $Lf=\sum_{i=0}^n f(x_i)\ell_i$.
We also know that
$$ L(af+bg)=\sum_{i=0}^n(af)(x_i)\ell_i+\sum_{i=0}^n(bg)(x_i)\ell_i=a\sum_{i=0}^n f(x_i)\ell_i+b\sum_{i=0}^n g(x_i)\ell_i=aLf+bLg $$
and thus $L$ is linear.
Is this reasoning correct?
There is a mapping $f\mapsto p$. Denote this mapping by $L$ and show that $Lf=\sum_{i=0}^n f(x_i)\ell_i$