[proofplan] The induced class function $\operatorname{Ind}^G_H \psi$ is by construction an element of $\mathcal{C}(G)$, so to show it is a character we must prove it is a non-negative-integer combination of irreducible characters of $G$. We pair $\operatorname{Ind}^G_H \psi$ with each irreducible character $\chi$ of $G$ in the $G$-inner product, and show $\langle \operatorname{Ind}^G_H \psi, \chi \rangle_G \in \mathbb{Z}_{\geq 0}$. The bridge is [Frobenius Reciprocity](/theorems/2449), which converts this $G$-pairing to an $H$-pairing $\langle \psi, \operatorname{Res}^G_H \chi \rangle_H$ — and the latter is a non-negative integer because $\psi$ and $\operatorname{Res}^G_H \chi$ are both genuine characters of $H$, and the inner product of two characters in the same group is the multiplicity in their direct-sum decomposition. [/proofplan] [step:Reduce the claim to showing $\langle \operatorname{Ind}^G_H \psi, \chi \rangle_G \in \mathbb{Z}_{\geq 0}$ for every irreducible character $\chi$ of $G$] Let $\chi_1, \ldots, \chi_k$ be the irreducible characters of $G$. By the [Number of Irreducible Characters](/theorems/2428) and [Row Orthogonality](/theorems/2430), $\{\chi_1, \ldots, \chi_k\}$ is an orthonormal basis of $\mathcal{C}(G)$ with respect to the $G$-inner product. The induced class function $\operatorname{Ind}^G_H \psi \in \mathcal{C}(G)$ therefore expands uniquely as \begin{align*} \operatorname{Ind}^G_H \psi = \sum_{i=1}^k a_i\, \chi_i, \qquad a_i = \langle \operatorname{Ind}^G_H \psi, \chi_i \rangle_G \in \mathbb{C}. \end{align*} By the [Characterisation of Characters via Inner Products](/theorems/2426) (criterion: a class function is a character iff it is a non-negative-integer combination of irreducibles), it suffices to show \begin{align*} a_i = \langle \operatorname{Ind}^G_H \psi, \chi_i \rangle_G \in \mathbb{Z}_{\geq 0} \qquad \text{for every } i. \end{align*} [/step] [step:Apply Frobenius reciprocity to convert each coefficient $a_i$ into an $H$-inner product] Fix an irreducible character $\chi = \chi_i$ of $G$. By [Frobenius Reciprocity](/theorems/2449), \begin{align*} \langle \operatorname{Ind}^G_H \psi, \chi \rangle_G = \langle \psi, \operatorname{Res}^G_H \chi \rangle_H. \end{align*} The hypotheses of Frobenius reciprocity are satisfied: $\psi \in \mathcal{C}(H)$ since $\psi$ is a character of $H$ (and characters are class functions), and $\chi \in \mathcal{C}(G)$ since $\chi$ is a character of $G$. It now remains to show $\langle \psi, \operatorname{Res}^G_H \chi \rangle_H \in \mathbb{Z}_{\geq 0}$. [/step] [step:Argue $\operatorname{Res}^G_H \chi$ is a character of $H$ and hence the $H$-inner product is a non-negative integer] The restriction $\operatorname{Res}^G_H \chi$ is the character of the restricted representation: if $\rho: G \to \operatorname{GL}(V)$ is a representation of $G$ with character $\chi$, then $\rho\big|_H: H \to \operatorname{GL}(V)$ is a representation of $H$ — call it $\rho^H$ — and its character is $\operatorname{Res}^G_H \chi$. So $\operatorname{Res}^G_H \chi$ is a character of $H$. By [Maschke's Theorem](/theorems/2409), the representation $\rho^H$ decomposes as a direct sum of irreducibles. Let $\psi_1, \ldots, \psi_\ell$ denote the irreducible characters of $H$. Then there exist non-negative integers $b_1, \ldots, b_\ell$ such that \begin{align*} \operatorname{Res}^G_H \chi = \sum_{j=1}^\ell b_j \psi_j, \qquad b_j \in \mathbb{Z}_{\geq 0}. \end{align*} Similarly, $\psi$ is a character of $H$ by hypothesis, and decomposes as \begin{align*} \psi = \sum_{j=1}^\ell c_j \psi_j, \qquad c_j \in \mathbb{Z}_{\geq 0}. \end{align*} The $H$-inner product, sesquilinear and with $\{\psi_j\}$ orthonormal by [Row Orthogonality](/theorems/2430) applied to $H$, gives \begin{align*} \langle \psi, \operatorname{Res}^G_H \chi \rangle_H = \biggl\langle \sum_j c_j \psi_j, \sum_{j'} b_{j'} \psi_{j'} \biggr\rangle_H = \sum_{j, j'} c_j\, \overline{b_{j'}}\, \delta_{jj'} = \sum_j c_j b_j. \end{align*} The product of non-negative integers is a non-negative integer, and a finite sum of non-negative integers is a non-negative integer: \begin{align*} \langle \psi, \operatorname{Res}^G_H \chi \rangle_H = \sum_{j=1}^\ell c_j b_j \in \mathbb{Z}_{\geq 0}. \end{align*} [guided] The argument is a direct application of the principle that the inner product of two characters in any group counts how many irreducible components they share. Concretely: **What is $\langle \psi, \operatorname{Res}^G_H \chi \rangle_H$ counting?** Geometrically, it counts the number of common irreducible $H$-factors in the direct-sum decompositions of $\psi$ and $\operatorname{Res}^G_H \chi$, weighted by their multiplicities. Algebraically, with $\psi = \sum_j c_j \psi_j$ and $\operatorname{Res}^G_H \chi = \sum_j b_j \psi_j$, the inner product picks out $\sum_j c_j b_j$ — the dot product of the multiplicity vectors. **Why are $b_j, c_j \in \mathbb{Z}_{\geq 0}$?** Because $\psi$ and $\operatorname{Res}^G_H \chi$ are genuine characters of $H$ — the characters of actual representations, not virtual differences. Genuine characters have non-negative integer multiplicities by definition (they are direct sums, not formal differences). The fact that $\operatorname{Res}^G_H \chi$ is a genuine character — not virtual — relies on $\chi$ being a character of an actual $G$-representation $\rho$, restricted to $H$. The restricted representation $\rho\big|_H$ is itself a genuine $H$-representation, and its character is $\operatorname{Res}^G_H \chi$. **Why is the conclusion that $\operatorname{Ind}^G_H \psi$ is a character?** Because being a character is precisely being a non-negative-integer combination of irreducibles. The $a_i = \langle \operatorname{Ind}^G_H \psi, \chi_i \rangle_G$ are the multiplicities in the irreducible-character expansion. Showing they are non-negative integers is the entire content of being a character. The [Characterisation of Characters via Inner Products](/theorems/2426) packages this equivalence into a usable criterion. **The Frobenius bridge is the heart of the argument.** Without Frobenius reciprocity, we would have to reason about $\operatorname{Ind}^G_H \psi$ directly — and its definition is a sum that gives no immediate reason to expect a character. Frobenius reciprocity converts the $G$-inner product (where $\operatorname{Ind}^G_H \psi$ is hard to handle) into an $H$-inner product (where $\psi$ and $\operatorname{Res}^G_H \chi$ are both transparently characters). The transfer is the proof. [/guided] [/step] [step:Conclude $\operatorname{Ind}^G_H \psi$ is a character] By Steps 2 and 3, every coefficient $a_i = \langle \operatorname{Ind}^G_H \psi, \chi_i \rangle_G$ in the expansion \begin{align*} \operatorname{Ind}^G_H \psi = \sum_{i=1}^k a_i\, \chi_i \end{align*} is a non-negative integer. By the [Characterisation of Characters via Inner Products](/theorems/2426), $\operatorname{Ind}^G_H \psi$ is a character of $G$. This completes the proof. [/step]