Notice that by your definition of norm, any Cauchy sequence will induce two Cauchy sequences in namely and . Since is complete this implies that there exists , such that and . Now it is a standard result that if is a sequence of functions such that the seguence of derivatives converges uniformly and converges in one point (you have uniform convergence for this!) then is and . It follows that your space is complete.
To show that a finite dim. normed space is comlete use the fact that all norms are equivalent.