Is there any distinction between these products: scalar, dot, inner?

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.

Read more: http://www.physicsforums.com/showthread.php?t=735158

Solution 1:

Scalar product, inner product and dot product should refer to the same thing, but the last one is normally only used in context of classical vector calculus or if your vector space is $\mathbb R^n$.

Applying a covector to a vector is a special case of tensor contraction that is known as the natural or duality pairing. In the language of differential forms, it is also a special case of the interior product.

The geometric algebra is the Clifford algebra induced by an inner product. As a set, it's the same as the exterior algebra of multivectors, but comes with an additional product - the 'dotless' one.

Solution 2:

Let $V$ be a complex vector space. Then the inner product will be a map: \begin{equation} \langle \;\; , \;\; \rangle : V \times V \to \mathbb{C} \end{equation} which is (1) conjugate symmetric in both arguments: \begin{equation} \langle \mathbf{u} , \mathbf{v} \rangle = \overline{\langle \mathbf{v} , \mathbf{u} \rangle} \end{equation} (2) linear in the second argument: \begin{equation} \langle \mathbf{u}, a \mathbf{v} \rangle = a \langle \mathbf{u},\mathbf{v} \rangle \end{equation} where $a$ is some constant, and: \begin{equation} \langle \mathbf{u} , \mathbf{v} + \mathbf{w} \rangle = \langle \mathbf{u} , \mathbf{v} \rangle + \langle \mathbf{u} , \mathbf{w} \rangle \end{equation} (3) the inner product is positive-definite: \begin{equation} \langle \mathbf{u} , \mathbf{u} \rangle \geq 0 \end{equation} and: \begin{equation} \langle \mathbf{u} , \mathbf{u} \rangle = 0 \iff \mathbf{u} = \mathbf{0} \end{equation} It important to note that there are two different conventions of condition (2). The one I used implies that: \begin{equation} \langle \mathbf{u}, a \mathbf{v} \rangle = a \langle \mathbf{u},\mathbf{v} \rangle \implies \langle a \mathbf{u},\mathbf{v} \rangle = \overline{\langle \mathbf{v} , a \mathbf{u} \rangle} = \overline{a \langle \mathbf{v} , \mathbf{u} \rangle} = \overline{a} \langle \mathbf{u},\mathbf{v} \rangle \end{equation}

Finally, note that if the vector spaces are real, then the complex conjugation has no effect and so the inner product is linear in the first and second argument. If the space is $\mathbb{R}^n$, then it becomes the dot product: \begin{equation} \langle \mathbf{u},\mathbf{v}\rangle = \mathbf{u} \cdot \mathbf{v} = \sum\limits_{i=1}^n u^i v^i \end{equation}

Edit: I should also mention that often it is useful to weaken the conditions on the inner product. Namely, to let the inner product not be positive-definite as is the case for the innner product in Minkowski space. I think that this is the inner product that Hestenes is referring to.