I hope you will forgive a math question that comes up in physics contexts where language is loose. This question migrated from Physics SE. I'm finding that I sometimes don't know what kind of product a physics author means. There is a product which can be defined on a vector space that takes two vectors and returns a scalar. There is a product that takes a vector and a covector and returns a scalar.
Is there an agreed-upon language for distinctions between the three words "scalar product", "dot product", and "inner product"?
I can mostly gloss over the language as it's usually clear from context. But not always.
Addition: I think we can all agree what a scalar product is: a map taking two vectors and returning a scalar. A vector space does not have to have a scalar product. To have a scalar product, a vector space needs a metric.
And we can all agree that there is a "contraction" taking one member of a vector space, and one member of it dual, and retuning a scalar, where the dual can be considered to be a scalar-valued linear function on the vector space. Does this "product" have a name?
What I would like to know: what are the definitions of "dot product" and "inner product"?
Addition #2: A commenter (@Hunter) cited a link that points out that physics authors do not all agree on what an inner product is. The following quotation is cited: "If the inner product is taken of two vectors, one must be a contravariant vector and the other a covariant vector. The inner product of two covariant or two contravariant vectors is not defined." [Spherical Astronomy, Robin Michael Green page 495.] Some say the inner product takes $V^*\times V$ into scalars, others say $V \times V$. (Consensus among physicists here is that "inner" = "scalar", i.e. the domain for both is $V \times V$., and "dot" = "scalar", usually reserved for Euclidean geometry.)
Furthermore, most quantum mechanics texts call $\langle\psi\mid\chi\rangle$ an inner product, whereas by my understanding this is a mapping of one bra and one ket to scalars: a contraction of a vector with its dual. I understand that there is an isomorphism in this case, so there is no ambiguity, but the terminology adds to confusion.
One thing is certain: some authors don't tell us which definition they are using, and it's sometimes not clear from context.