Bilinear map

{{Short description|Function of two vectors linear in each argument}}

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.

A bilinear map can also be defined for modules. For that, see the article pairing.

== 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 second entry of the bilinear map fixed while letting the first entry vary, yielding <math>B_w</math>, the result is a linear operator, and similarly for when we hold the first 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 {{nowrap|1=''B''(''v'', ''w'') = ''B''(''w'', ''v'')}} 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 {{nowrap|''B'' : ''M'' × ''N'' → ''T''}} with ''T'' an {{nowrap|(''R'', ''S'')}}-bimodule, and for which any ''n'' in ''N'', {{nowrap|''m'' ↦ ''B''(''m'', ''n'')}} is an ''R''-module homomorphism, and for any ''m'' in ''M'', {{nowrap|''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'') ⋅ ''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 {{nowrap|1=''B''(''v'', ''w'') = 0_''X''}} whenever {{nowrap|1=''v'' = 0_''V''}} or {{nowrap|1=''w'' = 0_''W''}}. This may be seen by writing the zero vector 0_''V'' as {{nowrap|0 ⋅ 0_''V''}} (and similarly for 0_''W'') and moving the scalar 0 "outside", in front of ''B'', by linearity.

The set {{nowrap|''L''(''V'', ''W''; ''X'')}} of all bilinear maps is a linear subspace of the space (viz. vector space, module) of all maps from {{nowrap|''V'' × ''W''}} into ''X''.

If ''V'', ''W'', ''X'' are finite-dimensional, then so is {{nowrap|''L''(''V'', ''W''; ''X'')}}. For <math>X = F,</math> that is, bilinear forms, the dimension of this space is {{nowrap|dim ''V'' × dim ''W''}} (while the space {{nowrap|''L''(''V'' × ''W''; ''F'')}} of ''linear'' forms is of dimension {{nowrap|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 {{nowrap|''B''(''e''_''i'', ''f''_''j'')}}, and vice versa. Now, if ''X'' is a space of higher dimension, we obviously have {{nowrap|1=dim ''L''(''V'', ''W''; ''X'') = dim ''V'' × dim ''W'' × dim ''X''}}.

== Examples == * Matrix multiplication is a bilinear map {{nowrap|M(''m'', ''n'') × M(''n'', ''p'') → M(''m'', ''p'')}}. * 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 {{nowrap|''V'' × ''V'' → ''F''}}. * If ''V'' is a vector space with dual space ''V''^∗, then the canonical evaluation map, {{nowrap|1=''b''(''f'', ''v'') = ''f''(''v'')}} is a bilinear map from {{nowrap|''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 {{nowrap|1=''b''(''v'', ''w'') = ''f''(''v'')''g''(''w'')}} defines a bilinear map {{nowrap|''V'' × ''W'' → ''F''}}. * 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 {{nowrap|(''v'', ''u'') ↦ ''B''(''v'', ''Lu'')}} is a bilinear map on {{nowrap|''V'' × ''U''}}.

== 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 '''{{visible anchor|separately continuous}}''' 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.{{sfn | Trèves | 2006 | pp=424-426}} 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.{{sfn | Trèves | 2006 | pp=424-426}} * 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.{{sfn | Trèves | 2006 | pp=424-426}} * If a bilinear map is continuous at (0, 0) then it is continuous everywhere.{{sfn | Schaefer|Wolff| 1999 | p=118}}

=== Composition map === {{See also|Topology of uniform convergence}}

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.{{sfn | Trèves | 2006 | pp=424-426}} * 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>{{sfn| Trèves | 2006 | pp=424-426}}

==See also==

* {{annotated link|Tensor product}} * {{annotated link|Sesquilinear form}} * {{annotated link|Bilinear filtering}} * {{annotated link|Multilinear map}}

==References==

{{reflist}} {{reflist|group=note}}

==Bibliography==

* {{Schaefer Wolff Topological Vector Spaces|edition=2}} * {{Trèves François Topological vector spaces, distributions and kernels}}

==External links== * {{springer|title=Bilinear mapping|id=p/b016280}}

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Category:Bilinear maps Category:Multilinear algebra