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.
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 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 (see for example scalar product, inner product and quadratic form).
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.
A first 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.
If V, W, X are finite-dimensional, then so is L(V, W; X). For X = F, i.e. 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.
- Matrix multiplication is a bilinear map 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 × V → 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 application operator, 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 R3 is a bilinear map R3 × R3 → R3.
- Let B : V × W → X be a bilinear map, and L : U → 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 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.
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 X, Y, and Z 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 X, Y, and Z be locally convex Hausdorff spaces and let be the composition map defined by . 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 then the restriction is continuous for all three topologies.
- If Y is a barreled space then for every sequence converging to u in and every sequence converging to v in , the sequence converges to in .