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
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
and we have B(v, w) = B(w, v) for all
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) = 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
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.
Continuity and separate continuity
Suppose
and
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.
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 to be continuous.
- If X is a Baire space and Y is metrizable then every separately continuous bilinear map
is continuous.
- If
are the strong duals of Fréchet spaces then every separately continuous bilinear map
is continuous.
- If a bilinear map is continuous at (0, 0) then it is continuous everywhere.
Composition map
Let
be locally convex Hausdorff spaces and let
be the composition map defined by
In general, the bilinear map
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
is an equicontinuous subset of
then the restriction
is continuous for all three topologies.
- If
is a barreled space then for every sequence
converging to
in
and every sequence
converging to
in
the sequence
converges to
in