# Bilinear map

Short description: Function of two vectors linear in each argument

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

### Vector spaces

Let $\displaystyle{ V, W }$ and $\displaystyle{ X }$ be three vector spaces over the same base field $\displaystyle{ F }$. A bilinear map is a function $\displaystyle{ B : V \times W \to X }$ such that for all $\displaystyle{ w \in W }$, the map $\displaystyle{ B_w }$ $\displaystyle{ v \mapsto B(v, w) }$ is a linear map from $\displaystyle{ V }$ to $\displaystyle{ X, }$ and for all $\displaystyle{ v \in V }$, the map $\displaystyle{ B_v }$ $\displaystyle{ w \mapsto B(v, w) }$ is a linear map from $\displaystyle{ W }$ to $\displaystyle{ X. }$ 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 $\displaystyle{ B }$ satisfies the following properties.

• For any $\displaystyle{ \lambda \in F }$, $\displaystyle{ B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). }$
• The map $\displaystyle{ B }$ is additive in both components: if $\displaystyle{ v_1, v_2 \in V }$ and $\displaystyle{ w_1, w_2 \in W, }$ then $\displaystyle{ B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w) }$ and $\displaystyle{ B(v, w_1 + w_2) = B(v, w_1) + B(v, w_2). }$

If $\displaystyle{ V = W }$ and we have B(v, w) = B(w, v) for all $\displaystyle{ v, w \in V, }$ 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 × NT with T an (R, S)-bimodule, and for which any n in N, mB(m, n) is an R-module homomorphism, and for any m in M, nB(m, n) is an S-module homomorphism. This satisfies

B(rm, n) = rB(m, n)
B(m, ns) = 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) = 0X whenever v = 0V or w = 0W. This may be seen by writing the zero vector 0V as 0 ⋅ 0V (and similarly for 0W) 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 $\displaystyle{ X = F, }$ 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(ei, fj), 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 $\displaystyle{ \R }$ carries an inner product, then the inner product is a bilinear map $\displaystyle{ V \times V \to \R. }$
• In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × VF.
• 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 × WF.
• The cross product in $\displaystyle{ \R^3 }$ is a bilinear map $\displaystyle{ \R^3 \times \R^3 \to \R^3. }$
• Let $\displaystyle{ B : V \times W \to X }$ be a bilinear map, and $\displaystyle{ L : U \to W }$ be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.

## Continuity and separate continuity

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

1. for all $\displaystyle{ x \in X, }$ the map $\displaystyle{ Y \to Z }$ given by $\displaystyle{ y \mapsto b(x, y) }$ is continuous;
2. for all $\displaystyle{ y \in Y, }$ the map $\displaystyle{ X \to Z }$ given by $\displaystyle{ x \mapsto b(x, y) }$ 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

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 $\displaystyle{ b : X \times Y \to Z }$ is continuous.[1]
• If $\displaystyle{ X, Y, \text{ and } Z }$ are the strong duals of Fréchet spaces then every separately continuous bilinear map $\displaystyle{ b : X \times Y \to Z }$ is continuous.[1]
• If a bilinear map is continuous at (0, 0) then it is continuous everywhere.[2]

### Composition map

Let $\displaystyle{ X, Y, \text{ and } Z }$ be locally convex Hausdorff spaces and let $\displaystyle{ C : L(X; Y) \times L(Y; Z) \to L(X; Z) }$ be the composition map defined by $\displaystyle{ C(u, v) := v \circ u. }$ In general, the bilinear map $\displaystyle{ C }$ 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:

1. give all three the topology of bounded convergence;
2. give all three the topology of compact convergence;
3. give all three the topology of pointwise convergence.
• If $\displaystyle{ E }$ is an equicontinuous subset of $\displaystyle{ L(Y; Z) }$ then the restriction $\displaystyle{ C\big\vert_{L(X; Y) \times E} : L(X; Y) \times E \to L(X; Z) }$ is continuous for all three topologies.[1]
• If $\displaystyle{ Y }$ is a barreled space then for every sequence $\displaystyle{ \left(u_i\right)_{i=1}^{\infty} }$ converging to $\displaystyle{ u }$ in $\displaystyle{ L(X; Y) }$ and every sequence $\displaystyle{ \left(v_i\right)_{i=1}^{\infty} }$ converging to $\displaystyle{ v }$ in $\displaystyle{ L(Y; Z), }$ the sequence $\displaystyle{ \left(v_i \circ u_i\right)_{i=1}^{\infty} }$ converges to $\displaystyle{ v \circ u }$ in $\displaystyle{ L(Y; Z). }$ [1]