Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 4 / 19F / c / Solution

Let $m=[G:H]$, extend $\psi$ to ${\psi }_0$ by zero outside $H$, and choose coset representatives $x_1,\dots ,x_m$. Then $\mathrm{Ind}\psi (g)={\sum }_{j=1}^m{f}_j(g)$ with $f_j(g)={\psi }_0(x_j^{-1}g{x}_j)$. Each summand has squared $G$-normsince $\psi$ is irreducible. Pointwise Cauchy-Schwarz inequality gives $|\sum f_j{|}^2\le m\sum |f_j{|}^2$. After summing and dividing by $|G|$,The $e_i$ are nonnegative integers by part (b). This proves the character induction norm bound without an assumption that $H$ is normal.