# Bilinear map

In mathematics, a **bilinear map** is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments. Matrix multiplication is an example.

## Definition[edit]

### Vector spaces[edit]

Let and be three vector spaces over the same base field . A bilinear map is a function

such that for all , the map

is a linear map from to , and for all , the map

is a linear map from to . In other words, when we hold the first entry of the bilinear map fixed while letting the second entry vary, the result is a linear operator, and similarly for when we hold the second entry fixed.

Such a map satisfies the following properties.

- For any , .
- The map is additive in both components: if and , then and .

If *V* = *W* and we have *B*(*v*, *w*) = *B*(*w*, *v*) for all *v*, *w* in *V*, then we say that *B* is *symmetric*. If *X* is the base field *F*, then the map is called a **bilinear form**, which are well-studied (see for example *Scalar product*, *Inner product* and *Quadratic form*).

### Modules[edit]

The definition works without any changes if instead of vector spaces over a field *F*, we use modules over a commutative ring *R*. It generalizes to *n*-ary functions, where the proper term is *multilinear*.

For non-commutative rings *R* and *S*, a left *R*-module *M* and a right *S*-module *N*, a bilinear map is a map *B* : *M* × *N* → *T* with *T* an (*R*, *S*)-bimodule, and for which any *n* in *N*, *m* ↦ *B*(*m*, *n*) is an *R*-module homomorphism, and for any *m* in *M*, *n* ↦ *B*(*m*, *n*) is an *S*-module homomorphism. This satisfies

*B*(*r*⋅*m*,*n*) =*r*⋅*B*(*m*,*n*)*B*(*m*,*n*⋅*s*) =*B*(*m*,*n*) ⋅*s*

for all *m* in *M*, *n* in *N*, *r* in *R* and *s* in *S*, as well as *B* being additive in each argument.

## Properties[edit]

An immediate consequence of the definition is that *B*(*v*, *w*) = 0_{X} whenever *v* = 0_{V} or *w* = 0_{W}. This may be seen by writing the zero vector 0_{V} as 0 ⋅ 0_{V} (and similarly for 0_{W}) and moving the scalar 0 "outside", in front of *B*, by linearity.

The set *L*(*V*, *W*; *X*) of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from *V* × *W* into *X*.

If *V*, *W*, *X* are finite-dimensional, then so is *L*(*V*, *W*; *X*). For *X* = *F*, i.e. bilinear forms, the dimension of this space is dim *V* × dim *W* (while the space *L*(*V* × *W*; *F*) of *linear* forms is of dimension dim *V* + dim *W*). To see this, choose a basis for *V* and *W*; then each bilinear map can be uniquely represented by the matrix *B*(*e*_{i}, *f*_{j}), and vice versa.
Now, if *X* is a space of higher dimension, we obviously have dim *L*(*V*, *W*; *X*) = dim *V* × dim *W* × dim *X*.

## Examples[edit]

- Matrix multiplication is a bilinear map M(
*m*,*n*) × M(*n*,*p*) → M(*m*,*p*). - If a vector space
*V*over the real numbers**R**carries an inner product, then the inner product is a bilinear map*V*×*V*→**R**. - In general, for a vector space
*V*over a field*F*, a bilinear form on*V*is the same as a bilinear map*V*×*V*→*F*. - If
*V*is a vector space with dual space*V*^{∗}, then the application operator,*b*(*f*,*v*) =*f*(*v*) is a bilinear map from*V*^{∗}×*V*to the base field. - Let
*V*and*W*be vector spaces over the same base field*F*. If*f*is a member of*V*^{∗}and*g*a member of*W*^{∗}, then*b*(*v*,*w*) =*f*(*v*)*g*(*w*) defines a bilinear map*V*×*W*→*F*. - The cross product in
**R**^{3}is a bilinear map**R**^{3}×**R**^{3}→**R**^{3}. - Let
*B*:*V*×*W*→*X*be a bilinear map, and*L*:*U*→*W*be a linear map, then (*v*,*u*) ↦*B*(*v*,*Lu*) is a bilinear map on*V*×*U*.

## Continuity and separate continuity[edit]

Suppose *X*, *Y*, and *Z* are topological vector spaces and let be a bilinear map.
Then *b* is said to be **separately continuous** if the following two conditions hold:

- for all , the map given by is continuous;
- for all , the map given by is continuous.

Many separately continuous bilinear that are not continuous satisfy an additional property: hypocontinuity.^{[1]}
All continuous bilinear maps are hypocontinuous.

### Sufficient conditions for continuity[edit]

Many bilinear maps that occur in practice are separately continuous but not all are continuous. We list here sufficient conditions for a separately continuous bilinear to be continuous.

- If
*X*is a Baire space and*Y*is metrizable then every separately continuous bilinear map is continuous.^{[1]} - If
*X*,*Y*, and*Z*are the strong duals of Fréchet spaces then every separately continuous bilinear map is continuous.^{[1]} - If a bilinear map is continuous at (0, 0) then it is continuous everywhere.
^{[2]}

### Composition map[edit]

Let *X*, *Y*, and *Z* be locally convex Hausdorff spaces and let be the composition map defined by .
In general, the bilinear map *C* is not continuous (no matter what topologies the spaces of linear maps are given).
We do, however, have the following results:

Give all three spaces of linear maps one of the following topologies:

- give all three the topology of bounded convergence;
- give all three the topology of compact convergence;
- give all three the topology of pointwise convergence.

- If
*E*is an equicontinuous subset of then the restriction is continuous for all three topologies.^{[1]} - If
*Y*is a barreled space then for every sequence converging to*u*in and every sequence converging to*v*in , the sequence converges to in .^{[1]}

## See also[edit]

## References[edit]

- ^
^{a}^{b}^{c}^{d}^{e}Trèves 2006, pp. 424-426. **^**Schaefer & Wolff 1999, p. 118.

## Bibliography[edit]

- Schaefer, Helmut H.; Wolff, Manfred P. (1999).
*Topological Vector Spaces*. GTM.**8**(Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. - Trèves, François (2006) [1967].
*Topological Vector Spaces, Distributions and Kernels*. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

## External links[edit]

- "Bilinear mapping",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994]