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 V , W {\displaystyle V,W} and X {\displaystyle X} be three vector spaces over the same base field F {\displaystyle F} . A bilinear map is a function
B : V × W → X {\displaystyle B:V\times W\to X} such that for all w ∈ W {\displaystyle w\in W} , the map B w {\displaystyle B_{w}} v ↦ B ( v , w ) {\displaystyle v\mapsto B(v,w)} is a linear map from V {\displaystyle V} to X , {\displaystyle X,} and for all v ∈ V {\displaystyle v\in V} , the map B v {\displaystyle B_{v}} w ↦ B ( v , w ) {\displaystyle w\mapsto B(v,w)} is a linear map from W {\displaystyle W} to X . {\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 B {\displaystyle B} satisfies the following properties.
If V = W {\displaystyle V=W} and we have B(v, w) = B(w, v) for all v , w ∈ V , {\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).
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) ⋅ sfor 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 X = F , {\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.
Suppose X , Y , {\displaystyle X,Y,} and Z {\displaystyle Z} are topological vector spaces and let b : X × Y → Z {\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:
Many separately continuous bilinear that are not continuous satisfy an additional property: hypocontinuity. All continuous bilinear maps are hypocontinuous.
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.
Let X , Y , and Z {\displaystyle X,Y,{\text{ and }}Z} be locally convex Hausdorff spaces and let C : L ( X ; Y ) × L ( Y ; Z ) → L ( X ; Z ) {\displaystyle C:L(X;Y)\times L(Y;Z)\to L(X;Z)} be the composition map defined by C ( u , v ) := v ∘ u . {\displaystyle C(u,v):=v\circ u.} In general, the bilinear map C {\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:
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