Inner product of functions

Bilinear form
This definition works in all non-modular characteristics and is bilinear.

Let $$G$$ be a finite group and $$k$$ be a field whose characteristic does not divide the order of $$G$$. Given two functions $$f_1,f_2:G \to k$$, define:

$$\langle f_1, f_2 \rangle_G = \frac{1}{|G|} \sum_{g \in G} f_1(g)f_2(g^{-1})$$

Note that $$1/|G|$$ makes sense as an element of $$k$$ because $$G$$ is finite and the characteristic of $$k$$ does not divide the order of $$G$$.

Hermitian inner product
This definition works over $$\mathbb{C}$$ or any subfield of $$\mathbb{C}$$ that is closed under complex conjugation.

Let $$k$$ be such a field and $$G$$ be a finite group. Given two functions $$f_1,f_2:G \to k$$, define:

$$\langle f_1, f_2 \rangle_G = \frac{1}{|G|} \sum_{g \in G} f_1(g)\overline{f_2(g)}$$

Note that this is a Hermitian positive-definite inner product.

Relation between the definitions
First, note that the two inner products defined are not the same thing over a field where both definitions are applicable. The former is bilinear while the latter is sesquilinear and positive-definite (these qualities make it Hermitian). However, the following is true:

For a character $$\chi$$ of a representation of a finite group $$G$$ and an element $$g \in G$$, $$\chi(g^{-1}) = \overline{\chi(g)}$$. See trace of inverse is complex conjugate of trace.

Thus, if we apply the inner product operation only to characters of representations, then both definitions work exactly the same way.