what is the difference between a Hermitian inner product and an inner product?

Question

Are there any difference? Or is the Hermitian inner product just a special case of an inner product on a complex space?

Answer 1

There's no substantive difference. I believe the reason for the terminology is that inner products are often (e.g. on Mathworld and here and here) introduced for real vector spaces, and in this context the Hermiticity condition is stated as a symmetry condition. With this definition, a Hermitian inner product is strictly speaking not an inner product, since it is not symmetric but Hermitian. This is not, however, due to symmetric inner products being useful for complex vector spaces; it's just to have a simple definition at a more elementary level without worrying about complex numbers.

The situtation is basically the same with symmetric matrices and Hermitian matrices. You could ask: "What's the difference between symmetric matrices and Hermitian matrices?" We have separate names for them not because symmetric complex matrices tend to be useful but because we don't want to bother everyone who deals with symmetric real matrices with complex conjugation when they don't need it.

So, to summarize: The most satisfying and economical viewpoint is that there is a single concept here, and Hermiticity specializes to symmetry in the case of real vector spaces; but because we don't want to introduce complex numbers in a context where they're not needed, the concept of an "inner product" is often defined via symmetry without reference to Hermiticity, and in that case the Hermitian inner product needs to be defined separately, since it's generally not symmetric.