First of all, I have to prove that for any function $u \in$ $W^{1,p}(a,b)$ there is a unique affine function $ v \in E = \{Ax+B; A,B \in \mathbb{R} \} $ with the same border conditions $ u(a)=v(a) $ and $ u(b)=v(b) $. Which I have solved making a system and seeing that it has only one solution. Then I have to deduce that $W^{1,p}(a,b)$ as direct sum of $ W^{1,p}_0(a,b) \oplus E$ where $E$ are affine functions, and get the respective projections, but I don't really know how to assure that any function can be expressed as a sum of such subspaces .
Thank you in advance .