Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ?
My idea is use fact that there is no complete norm on $X$. I tried to consider $A:=Y/\ker\varphi$, where $\varphi:Y\rightarrow X$ is a continuous map from some $F$-space $Y$. Then $A$ is also $F$-space and I tried to use homomorphism theorem. Is it good idea or is there any simpler way to do it ?