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.
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).
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.
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.
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