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.

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 {{nowrap|1=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 {{nowrap|B : M × N → T}} with T an {{nowrap|(R, S)}}-bimodule, and for which any n in N, {{nowrap|m ↦ B(m, n)}} is an R-module homomorphism, and for any m in M, {{nowrap|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 {{nowrap|1=B(v, w) = 0_X}} whenever {{nowrap|1=v = 0_V}} or {{nowrap|1=w = 0_W}}. This may be seen by writing the zero vector 0_V as {{nowrap|0 ⋅ 0_V}} (and similarly for 0_W) and moving the scalar 0 "outside", in front of B, by linearity.

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

If V, W, X are finite-dimensional, then so is {{nowrap|L(V, W; X)}}. For $X = F,$ that is, bilinear forms, the dimension of this space is {{nowrap|dim V × dim W}} (while the space {{nowrap|L(V × W; F)}} of linear forms is of dimension {{nowrap|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 {{nowrap|B(e_i, f_j)}}, and vice versa.

Now, if X is a space of higher dimension, we obviously have {{nowrap|1=dim L(V, W; X) = dim V × dim W × dim X}}.

Examples

Matrix multiplication is a bilinear map {{nowrap|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 $V \times V \to \R.$
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map {{nowrap|V × V → F}}.
If V is a vector space with dual space V^∗, then the canonical evaluation map, {{nowrap|1=b(f, v) = f(v)}} is a bilinear map from {{nowrap|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 {{nowrap|1=b(v, w) = f(v)g(w)}} defines a bilinear map {{nowrap|V × W → F}}.
The cross product in $\R^3$ is a bilinear map $\R^3 \times \R^3 \to \R^3.$
Let $B : V \times W \to X$ be a bilinear map, and $L : U \to W$ be a linear map, then {{nowrap|(v, u) ↦ B(v, Lu)}} is a bilinear map on {{nowrap|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 {{visible anchor|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.{{sfn | Trèves | 2006 | pp=424-426}}

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.{{sfn | Trèves | 2006 | pp=424-426}}
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.{{sfn | Trèves | 2006 | pp=424-426}}
If a bilinear map is continuous at (0, 0) then it is continuous everywhere.{{sfn | Schaefer|Wolff| 1999 | p=118}}

= 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.{{sfn | Trèves | 2006 | pp=424-426}}
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 $L(Y; Z).$ {{sfn| Trèves | 2006 | pp=424-426}}

References

Bibliography

{{Schaefer Wolff Topological Vector Spaces|edition=2}}
{{Trèves François Topological vector spaces, distributions and kernels}}

External links

{{springer|title=Bilinear mapping|id=p/b016280}}

Category:Bilinear maps

Category:Multilinear algebra