Английская Википедия: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 <math>V, W </math> and <math>X</math> be three vector spaces over the same base field <math>F</math>. A bilinear map is a function <math display=block>B : V \times W \to X</math> such that for all <math>w \in W</math>, the map <math>B_w</math> <math display=block>v \mapsto B(v, w)</math> is a linear map from <math>V</math> to <math>X,</math> and for all <math>v \in V</math>, the map <math>B_v</math> <math display=block>w \mapsto B(v, w)</math> is a linear map from <math>W</math> to <math>X.</math> 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 <math>B</math> satisfies the following properties.
- For any <math>\lambda \in F</math>, <math>B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w).</math>
- The map <math>B</math> is additive in both components: if <math>v_1, v_2 \in V</math> and <math>w_1, w_2 \in W,</math> then <math>B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w)</math> and <math>B(v, w_1 + w_2) = B(v, w_1) + B(v, w_2).</math>
If <math>V = W</math> and we have Шаблон:Nowrap for all <math>v, w \in V,</math> 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 Шаблон:Nowrap with T an Шаблон:Nowrap-bimodule, and for which any n in N, Шаблон:Nowrap is an R-module homomorphism, and for any m in M, Шаблон:Nowrap 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 Шаблон:Nowrap whenever Шаблон:Nowrap or Шаблон:Nowrap. This may be seen by writing the zero vector 0V as Шаблон:Nowrap (and similarly for 0W) and moving the scalar 0 "outside", in front of B, by linearity.
The set Шаблон:Nowrap of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from Шаблон:Nowrap into X.
If V, W, X are finite-dimensional, then so is Шаблон:Nowrap. For <math>X = F,</math> that is, bilinear forms, the dimension of this space is Шаблон:Nowrap (while the space Шаблон:Nowrap of linear forms is of dimension Шаблон:Nowrap). To see this, choose a basis for V and W; then each bilinear map can be uniquely represented by the matrix Шаблон:Nowrap, and vice versa. Now, if X is a space of higher dimension, we obviously have Шаблон:Nowrap.
Examples
- Matrix multiplication is a bilinear map Шаблон:Nowrap.
- If a vector space V over the real numbers <math>\R</math> carries an inner product, then the inner product is a bilinear map <math>V \times V \to \R.</math>
- In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map Шаблон:Nowrap.
- If V is a vector space with dual space V∗, then the canonical evaluation map, Шаблон:Nowrap is a bilinear map from Шаблон:Nowrap 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 Шаблон:Nowrap defines a bilinear map Шаблон:Nowrap.
- The cross product in <math>\R^3</math> is a bilinear map <math>\R^3 \times \R^3 \to \R^3.</math>
- Let <math>B : V \times W \to X</math> be a bilinear map, and <math>L : U \to W</math> be a linear map, then Шаблон:Nowrap is a bilinear map on Шаблон:Nowrap.
Continuity and separate continuity
Suppose <math>X, Y,</math> and <math>Z</math> are topological vector spaces and let <math>b : X \times Y \to Z</math> be a bilinear map. Then b is said to be Шаблон:Visible anchor if the following two conditions hold:
- for all <math>x \in X,</math> the map <math>Y \to Z</math> given by <math>y \mapsto b(x, y)</math> is continuous;
- for all <math>y \in Y,</math> the map <math>X \to Z</math> given by <math>x \mapsto b(x, y)</math> is continuous.
Many separately continuous bilinear that are not continuous satisfy an additional property: hypocontinuity.Шаблон:Sfn 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 map to be continuous.
- If X is a Baire space and Y is metrizable then every separately continuous bilinear map <math>b : X \times Y \to Z</math> is continuous.Шаблон:Sfn
- If <math>X, Y, \text{ and } Z</math> are the strong duals of Fréchet spaces then every separately continuous bilinear map <math>b : X \times Y \to Z</math> is continuous.Шаблон:Sfn
- If a bilinear map is continuous at (0, 0) then it is continuous everywhere.Шаблон:Sfn
Composition map
Let <math>X, Y, \text{ and } Z</math> be locally convex Hausdorff spaces and let <math>C : L(X; Y) \times L(Y; Z) \to L(X; Z)</math> be the composition map defined by <math>C(u, v) := v \circ u.</math> In general, the bilinear map <math>C</math> 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 <math>E</math> is an equicontinuous subset of <math>L(Y; Z)</math> then the restriction <math>C\big\vert_{L(X; Y) \times E} : L(X; Y) \times E \to L(X; Z)</math> is continuous for all three topologies.Шаблон:Sfn
- If <math>Y</math> is a barreled space then for every sequence <math>\left(u_i\right)_{i=1}^{\infty}</math> converging to <math>u</math> in <math>L(X; Y)</math> and every sequence <math>\left(v_i\right)_{i=1}^{\infty}</math> converging to <math>v</math> in <math>L(Y; Z),</math> the sequence <math>\left(v_i \circ u_i\right)_{i=1}^{\infty}</math> converges to <math>v \circ u</math> in <math>L(Y; Z).</math> Шаблон:Sfn
See also
References
Bibliography
- Шаблон:Schaefer Wolff Topological Vector Spaces
- Шаблон:Trèves François Topological vector spaces, distributions and kernels
External links
Шаблон:Functional Analysis Шаблон:Authority control
