Inner products with more than two entries

Question

Is there a generalization for inner products for more than two entries, for instance $\langle x,y,z\rangle = \sum_{i\in I}x_iy_iz_i$? Of course such a product could lose symmetry, $\langle x,y,z\rangle \neq \langle z,x,y\rangle$ so there might be several ways to generalize this notion, but what this called?

Answer 1

On a vector space $\Bbb V$ over a field $\Bbb F$, a multilinear map $$\underbrace{\Bbb V \times \cdots \times \Bbb V}_r \to \Bbb F$$ is a multilinear form (of rank $r$). In the special case $r = 2$, we use the special (probably familiar) term bilinear form, and for $r = 3$ you sometimes see the adjective trilinear used.

By the universal property, we can canonically extend any such form to a linear map $$\underbrace{\Bbb V \otimes \cdots \otimes \Bbb V}_r \to \Bbb F,$$ that is, as an element of $(\underbrace{\Bbb V \otimes \cdots \otimes \Bbb V}_r)^* \cong \underbrace{\Bbb V^* \otimes \cdots \otimes \Bbb V^*}_r$; we call an element of this latter tensor product a (covariant) tensor (of rank $r$) on $\Bbb V$.

If we require that the multilinear map be symmetric, we use the terms symmetric multilinear form (and so for appropriate ranks symmetric bilinear form and symmetric trilinear form) and symmetric (covariant) tensor. If instead we require it be skew, we often use the term $r$-form.