Bilinear map

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 formula_1 and formula_2 be three vector spaces over the same base field formula_3. A bilinear map is a function
such that for all formula_5, the map formula_6
is a linear map from formula_8 to formula_2, and for all formula_10, the map formula_11
is a linear map from formula_13 to formula_2. 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 formula_15 satisfies the following properties.
If and we have 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").
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 with "T" an -bimodule, and for which any "n" in "N", is an "R"-module homomorphism, and for any "m" in "M", is an "S"-module homomorphism. This satisfies
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 whenever or . This may be seen by writing the zero vector 0"V" as (and similarly for 0"W") and moving the scalar 0 "outside", in front of "B", by linearity.
The set of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from into "X".
If "V", "W", "X" are finite-dimensional, then so is . For , i.e. bilinear forms, the dimension of this space is (while the space of "linear" forms is of dimension ). To see this, choose a basis for "V" and "W"; then each bilinear map can be uniquely represented by the matrix , and vice versa. 
Now, if "X" is a space of higher dimension, we obviously have .
Continuity and separate continuity.
Suppose "X", "Y", and "Z" are topological vector spaces and let formula_23 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.
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.
Composition map.
Let "X", "Y", and "Z" be locally convex Hausdorff spaces and let formula_32 be the composition map defined by formula_33. 
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: