Bilinear map

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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[edit]

Vector spaces[edit]

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).

Modules[edit]

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.

Properties[edit]

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.

Examples[edit]

  • 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 continuity[edit]

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 continuity[edit]

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 map[edit]

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 also[edit]

External links[edit]

  • Hazewinkel, Michiel, ed. (2001) [1994], "Bilinear mapping", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
  1. ^ a b c d e Treves 2006, pp. 424-426.
  2. ^ Schaefer 1999, p. 118.