Let $\displaystyle X$ be the set of all polynomial functions of degree 4 or less on $\displaystyle [0,1]$. If $\displaystyle f=a_0+a_1x+a_2x^2+a_3x^3+a_4x^4$ and $\displaystyle g=b_0+b_1x+b_2x^2+b_3x^3+b_4x^4$, we define $\displaystyle d_2(f(x),g(x))=\sum_{n=0}^4 \mid a_n-b_n \mid$.

Further, we define $\displaystyle \rho_2(f(x),g(x))=\frac{d_2(f(x),g(x))}{1+d_2(f(x) ,g(x))}$.

Will $\displaystyle (X,\rho_2)$ be a complete metric space?