- Source: 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.
A bilinear map can also be defined for modules. For that, see the article pairing.
Definition
= Vector spaces
=Let
V
,
W
{\displaystyle V,W}
and
X
{\displaystyle X}
be three vector spaces over the same base field
F
{\displaystyle F}
. A bilinear map is a function
B
:
V
×
W
→
X
{\displaystyle B:V\times W\to X}
such that for all
w
∈
W
{\displaystyle w\in W}
, the map
B
w
{\displaystyle B_{w}}
v
↦
B
(
v
,
w
)
{\displaystyle v\mapsto B(v,w)}
is a linear map from
V
{\displaystyle V}
to
X
,
{\displaystyle X,}
and for all
v
∈
V
{\displaystyle v\in V}
, the map
B
v
{\displaystyle B_{v}}
w
↦
B
(
v
,
w
)
{\displaystyle w\mapsto B(v,w)}
is a linear map from
W
{\displaystyle W}
to
X
.
{\displaystyle 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
{\displaystyle B}
satisfies the following properties.
For any
λ
∈
F
{\displaystyle \lambda \in F}
,
B
(
λ
v
,
w
)
=
B
(
v
,
λ
w
)
=
λ
B
(
v
,
w
)
.
{\displaystyle B(\lambda v,w)=B(v,\lambda w)=\lambda B(v,w).}
The map
B
{\displaystyle B}
is additive in both components: if
v
1
,
v
2
∈
V
{\displaystyle v_{1},v_{2}\in V}
and
w
1
,
w
2
∈
W
,
{\displaystyle w_{1},w_{2}\in W,}
then
B
(
v
1
+
v
2
,
w
)
=
B
(
v
1
,
w
)
+
B
(
v
2
,
w
)
{\displaystyle 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
)
.
{\displaystyle B(v,w_{1}+w_{2})=B(v,w_{1})+B(v,w_{2}).}
If
V
=
W
{\displaystyle V=W}
and we have B(v, w) = B(w, v) for all
v
,
w
∈
V
,
{\displaystyle 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 (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 B : M × N → T with T an (R, S)-bimodule, and for which any n in N, m ↦ B(m, n) is an R-module homomorphism, and for any m in M, 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 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.
If V, W, X are finite-dimensional, then so is L(V, W; X). For
X
=
F
,
{\displaystyle X=F,}
that is, 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
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
{\displaystyle \mathbb {R} }
carries an inner product, then the inner product is a bilinear map
V
×
V
→
R
.
{\displaystyle V\times V\to \mathbb {R} .}
In general, for a vector space V over a field F, a bilinear form on V is the same as a bilinear map V × V → F.
If V is a vector space with dual space V∗, then the canonical evaluation map, 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 × W → F.
The cross product in
R
3
{\displaystyle \mathbb {R} ^{3}}
is a bilinear map
R
3
×
R
3
→
R
3
.
{\displaystyle \mathbb {R} ^{3}\times \mathbb {R} ^{3}\to \mathbb {R} ^{3}.}
Let
B
:
V
×
W
→
X
{\displaystyle B:V\times W\to X}
be a bilinear map, and
L
:
U
→
W
{\displaystyle L:U\to W}
be a linear map, then (v, u) ↦ B(v, Lu) is a bilinear map on V × U.
Continuity and separate continuity
Suppose
X
,
Y
,
{\displaystyle X,Y,}
and
Z
{\displaystyle Z}
are topological vector spaces and let
b
:
X
×
Y
→
Z
{\displaystyle b:X\times Y\to Z}
be a bilinear map.
Then b is said to be separately continuous if the following two conditions hold:
for all
x
∈
X
,
{\displaystyle x\in X,}
the map
Y
→
Z
{\displaystyle Y\to Z}
given by
y
↦
b
(
x
,
y
)
{\displaystyle y\mapsto b(x,y)}
is continuous;
for all
y
∈
Y
,
{\displaystyle y\in Y,}
the map
X
→
Z
{\displaystyle X\to Z}
given by
x
↦
b
(
x
,
y
)
{\displaystyle x\mapsto b(x,y)}
is continuous.
Many separately continuous bilinear that are not continuous satisfy an additional property: hypocontinuity.
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
b
:
X
×
Y
→
Z
{\displaystyle b:X\times Y\to Z}
is continuous.
If
X
,
Y
,
and
Z
{\displaystyle X,Y,{\text{ and }}Z}
are the strong duals of Fréchet spaces then every separately continuous bilinear map
b
:
X
×
Y
→
Z
{\displaystyle 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
,
and
Z
{\displaystyle X,Y,{\text{ and }}Z}
be locally convex Hausdorff spaces and let
C
:
L
(
X
;
Y
)
×
L
(
Y
;
Z
)
→
L
(
X
;
Z
)
{\displaystyle C:L(X;Y)\times L(Y;Z)\to L(X;Z)}
be the composition map defined by
C
(
u
,
v
)
:=
v
∘
u
.
{\displaystyle C(u,v):=v\circ u.}
In general, the bilinear map
C
{\displaystyle 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
{\displaystyle E}
is an equicontinuous subset of
L
(
Y
;
Z
)
{\displaystyle L(Y;Z)}
then the restriction
C
|
L
(
X
;
Y
)
×
E
:
L
(
X
;
Y
)
×
E
→
L
(
X
;
Z
)
{\displaystyle C{\big \vert }_{L(X;Y)\times E}:L(X;Y)\times E\to L(X;Z)}
is continuous for all three topologies.
If
Y
{\displaystyle Y}
is a barreled space then for every sequence
(
u
i
)
i
=
1
∞
{\displaystyle \left(u_{i}\right)_{i=1}^{\infty }}
converging to
u
{\displaystyle u}
in
L
(
X
;
Y
)
{\displaystyle L(X;Y)}
and every sequence
(
v
i
)
i
=
1
∞
{\displaystyle \left(v_{i}\right)_{i=1}^{\infty }}
converging to
v
{\displaystyle v}
in
L
(
Y
;
Z
)
,
{\displaystyle L(Y;Z),}
the sequence
(
v
i
∘
u
i
)
i
=
1
∞
{\displaystyle \left(v_{i}\circ u_{i}\right)_{i=1}^{\infty }}
converges to
v
∘
u
{\displaystyle v\circ u}
in
L
(
Y
;
Z
)
.
{\displaystyle L(Y;Z).}
See also
Tensor product – Mathematical operation on vector spaces
Sesquilinear form – Generalization of a bilinear form
Bilinear filtering – Method of interpolating functions on a 2D gridPages displaying short descriptions of redirect targets
Multilinear map – Vector-valued function of multiple vectors, linear in each argument
References
Bibliography
Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 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 links
"Bilinear mapping", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Kata Kunci Pencarian:
- Produk Tensor
- Terence Tao
- Bilinear map
- Bilinear form
- Symmetric bilinear form
- Bilinear
- Degenerate bilinear form
- Hypocontinuous bilinear map
- Tensor product
- Bilinear transformation
- Trace (linear algebra)
- Pairing
