Past exam of the mathematics course of the University of Cambridge / 2007 / iii / Paper 32 / 1 / Solution

Put $V={\mathbb{R}}^d$ and write $a^{(k)}$ for the degree-$k$ component of a tensor. We use the truncated tensor algebraHere $V^{\otimes 0}=\mathbb{R}$, and products of degree above $N$ are discarded. The answers to the three printed parts follow with these conventions; the middle part is missing from the converted TeX.