# 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.
A bilinear map can also be defined for modules. For that, see the article pairing.

## Definition
### Vector spaces

Let $V,W$ and $X$ be three vector spaces over the same base field $F$. A bilinear map is a function $B:V\times W\to X$ such that for all $w\in W$, the map $B_{w}$ $v\mapsto B(v,w)$ is a linear map from $V$ to $X,$ and for all $v\in V$, the map $B_{v}$ $w\mapsto B(v,w)$ is a linear map from $W$ to $X.$ 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 $B$ satisfies the following properties.
* For any $\lambda \in F$, $B(\lambda v,w)=B(v,\lambda w)=\lambda B(v,w).$
* The map $B$ is additive in both components: if $v_{1},v_{2}\in V$ and $w_{1},w_{2}\in W,$ then $B(v_{1}+v_{2},w)=B(v_{1},w)+B(v_{2},w)$ and $B(v,w_{1}+w_{2})=B(v,w_{1})+B(v,w_{2}).$
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 (for example: scalar product, inner product, and quadratic form).

### Modules

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

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 $X=F,$ that is, 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

* Matrix multiplication is a bilinear map M(m, n) × M(n, p) → M(m, p).
* If a vector space V over the real numbers $\mathbb {R}$ carries an inner product, then the inner product is a bilinear map $V\times V\to \mathbb {R} .$
* 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 canonical evaluation map, 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 $\mathbb {R} ^{3}$ is a bilinear map $\mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ^{3}.$
* Let $B:V\times W\to X$ be a bilinear map, and $L:U\to W$ be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.

## Continuity and separate continuity

Suppose $X,Y,$ and $Z$ are topological vector spaces and let $b:X\times Y\to Z$ be a bilinear map. Then b is said to be separately continuous if the following two conditions hold:
* for all $x\in X,$ the map $Y\to Z$ given by $y\mapsto b(x,y)$ is continuous;
* for all $y\in Y,$ the map $X\to Z$ given by $x\mapsto b(x,y)$ is continuous.
Many separately continuous bilinear that are not continuous satisfy an additional property: hypocontinuity. All continuous bilinear maps are hypocontinuous.

### Sufficient conditions for continuity

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 map to be continuous.
* If X is a Baire space and Y is metrizable then every separately continuous bilinear map $b:X\times Y\to Z$ is continuous.
* If $X,Y,{\text{ and }}Z$ are the strong duals of Fréchet spaces then every separately continuous bilinear map $b:X\times Y\to Z$ is continuous.
* If a bilinear map is continuous at (0, 0) then it is continuous everywhere.

### Composition map

Let $X,Y,{\text{ and }}Z$ be locally convex Hausdorff spaces and let $C:L(X;Y)\times L(Y;Z)\to L(X;Z)$ be the composition map defined by $C(u,v):=v\circ u.$ 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 $L(Y;Z)$ then the restriction $C{\big \vert }_{L(X;Y)\times E}:L(X;Y)\times E\to L(X;Z)$ is continuous for all three topologies.
* If $Y$ is a barreled space then for every sequence $\left(u_{i}\right)_{i=1}^{\infty }$ converging to $u$ in $L(X;Y)$ and every sequence $\left(v_{i}\right)_{i=1}^{\infty }$ converging to $v$ in $L(Y;Z),$ the sequence $\left(v_{i}\circ u_{i}\right)_{i=1}^{\infty }$ converges to $v\circ u$ in $L(Y;Z).$