Bilinear map

  (Redirected from Bilinear operator)

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.

DefinitionEdit

Vector spacesEdit

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 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).

ModulesEdit

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 × NT with T an (R, S)-bimodule, and for which any n in N, mB(m, n) is an R-module homomorphism, and for any m in M, nB(m, n) is an S-module homomorphism. This satisfies

B(rm, n) = rB(m, n)
B(m, ns) = 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.

PropertiesEdit

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.

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.

A matrix M determines a bilinear map into the reals by means of a real bilinear form (v, w) ↦ vMw, then associates of this are taken to the other three possibilities using duality and the musical isomorphism:

 

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.

ExamplesEdit

  • 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 × VR.
  • In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × VF.
  • 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 × WF.
  • The cross product in R3 is a bilinear map R3 × R3R3.
  • Let B : V × WX be a bilinear map, and L : UW be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.

Continuity and separate continuityEdit

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:

  1. for all  , the map   given by   is continuous;
  2. for all  , the map   given by   is continuous.

Many separately continuous bilinear that are not continuous satisfy an additional property: hypocontinuity.[1] All continuous bilinear maps are hypocontinuous.

Sufficient conditions for continuityEdit

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.[1]
  • If X, Y, and Z are the strong duals of Fréchet spaces then every separately continuous bilinear map   is continuous.[1]
  • If a bilinear map is continuous at (0, 0) then it is continuous everywhere.[2]

Composition mapEdit

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:

  1. give all three the topology of bounded convergence;
  2. give all three the topology of compact convergence;
  3. give all three the topology of pointwise convergence.
  • If E is an equicontinuous subset of   then the restriction   is continuous for all three topologies.[1]
  • 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  . [1]

See alsoEdit

ReferencesEdit

  1. ^ a b c d e Trèves 2006, pp. 424-426.
  2. ^ Schaefer & Wolff 1999, p. 118.

BibliographyEdit

  • Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
  • Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322.

External linksEdit