separately continuous
   HOME

TheInfoList



OR:

In mathematics, a bilinear map is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
combining elements of two
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s to yield an element of a third vector space, and is
linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...
in each of its arguments.
Matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
is an example.


Definition


Vector spaces

Let V, W and X be three
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over the same base
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
F. A bilinear map is a
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-oriente ...
B : V \times W \to X such that for all w \in W, the map B_w v \mapsto B(v, w) is a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
from V to X, and for all v \in V, the map B_v w \mapsto B(v, w) is a linear map from W to X. 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 B satisfies the following properties. * For any \lambda \in F, B(\lambda v,w) = B(v, \lambda w) = \lambda B(v, w). * The map B is additive in both components: if v_1, v_2 \in V and w_1, w_2 \in W, then B(v_1 + v_2, w) = B(v_1, w) + B(v_2, w) and B(v, w_1 + w_2) = B(v, w_1) + B(v, w_2). If V = W and we have for all v, w \in V, then we say that ''B'' is ''
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
''. If ''X'' is the base field ''F'', then the map is called a '' bilinear form'', which are well-studied (for example:
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alge ...
,
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, and quadratic form).


Modules

The definition works without any changes if instead of vector spaces over a field ''F'', we use
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
s 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 In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, in t ...
, 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 :''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 Additive may refer to: Mathematics * Additive function, a function in number theory * Additive map, a function that preserves the addition operation * Additive set-functionn see Sigma additivity * Additive category, a preadditive category with f ...
in each argument.


Properties

An immediate consequence of the definition is that whenever or . This may be seen by writing the
zero vector In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive iden ...
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. The abbreviation ''viz.'' (or ''viz'' without a full stop) is short for the Latin , which itself is a contraction of the Latin phrase ''videre licet'', meaning "it is permitted to see". It is used as a synonym for "namely", "that is to say", "to ...
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
,
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
) of all maps from into ''X''. If ''V'', ''W'', ''X'' are finite-dimensional, then so is . For X = F, that is, bilinear forms, the dimension of this space is (while the space of ''linear'' forms is of dimension ). To see this, choose a
basis Basis may refer to: Finance and accounting * Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates * Basis trading, a trading strategy consisting ...
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 .


Examples

*
Matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
is a bilinear map . * If a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
''V'' over the
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s \R carries an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
, then the inner product is a bilinear map V \times V \to \R. The product vector space has one dimension. * In general, for a vector space ''V'' over a field ''F'', a bilinear form on ''V'' is the same as a bilinear map . * If ''V'' is a vector space with dual space ''V'', then the application operator, is a bilinear map from 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 defines a bilinear map . * The cross product in \R^3 is a bilinear map \R^3 \times \R^3 \to \R^3. * Let B : V \times W \to X be a bilinear map, and L : U \to W be a
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
, then is a bilinear map on .


Continuity and separate continuity

Suppose X, Y, \text Z are
topological vector space In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis. A topological vector space is a vector space that is als ...
s and let b : X \times Y \to Z be a bilinear map. Then ''b'' is said to be if the following two conditions hold: # for all x \in X, the map Y \to Z given by y \mapsto b(x, y) is continuous; # for all y \in Y, the map X \to Z given by x \mapsto b(x, y) is continuous. Many separately continuous bilinear that are not continuous satisfy an additional property:
hypocontinuity In mathematics, a hypocontinuous is a condition on bilinear maps of topological vector spaces that is weaker than continuity but stronger than separate continuity. Many important bilinear maps that are not continuous are, in fact, hypocontinuous. ...
. 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. * If ''X'' is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are e ...
and ''Y'' is
metrizable In topology and related areas of mathematics, a metrizable space is a topological space that is homeomorphic to a metric space. That is, a topological space (X, \mathcal) is said to be metrizable if there is a metric d : X \times X \to , \infty) s ...
then every separately continuous bilinear map b : X \times Y \to Z is continuous. * If X, Y, \text Z are the
strong dual In functional analysis and related areas of mathematics, the strong dual space of a topological vector space (TVS) X is the continuous dual space X^ of X equipped with the strong (dual) topology or the topology of uniform convergence on bounded sub ...
s of
Fréchet space In functional analysis and related areas of mathematics, Fréchet spaces, named after Maurice Fréchet, are special topological vector spaces. They are generalizations of Banach spaces ( normed vector spaces that are complete with respect to th ...
s then every separately continuous bilinear map b : X \times Y \to Z is continuous. * If a bilinear map is continuous at (0, 0) then it is continuous everywhere.


Composition map

Let X, Y, \text Z be locally convex Hausdorff spaces and let C : L(X; Y) \times L(Y; Z) \to L(X; Z) be the composition map defined by C(u, v) := v \circ u. 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: # 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 E is an
equicontinuous In mathematical analysis, a family of functions is equicontinuous if all the functions are continuous and they have equal variation over a given neighbourhood, in a precise sense described herein. In particular, the concept applies to countable fa ...
subset of L(Y; Z) then the restriction C\big\vert_ : L(X; Y) \times E \to L(X; Z) is continuous for all three topologies. * If Y is a barreled space then for every sequence \left(u_i\right)_^ converging to u in L(X; Y) and every sequence \left(v_i\right)_^ converging to v in L(Y; Z), the sequence \left(v_i \circ u_i\right)_^ converges to v \circ u in L(Y; Z).


See also

* * * *


References


Bibliography

* *


External links

* {{DEFAULTSORT:Bilinear Map Bilinear maps Multilinear algebra