 Character induction norm bound

Extend a class function $\psi$ on $H$ by zero outside $H$. For coset representatives $x_1,\dots ,x_m$ of $G/H$, ${\mathrm{Ind}}_H^G\psi (g)={\sum }_j{\psi }_0(x_j^{-1}g{x}_j)$. Each summand has squared normalized $G$-norm $\Vert \psi {\Vert }_H^2/m$. Pointwise Cauchy-Schwarz inequality and summation give $\Vert \mathrm{Ind}\psi {\Vert }_G^2\le m\Vert \psi {\Vert }_H^2$. If $\psi$ is an irreducible character, its norm is one, and the sum of squared induction multiplicities is at most the subgroup index.