Inmathematics, abilinear map is afunction combining elements of twovector spaces to yield an element of a third vector space, and islinear in each of its arguments.Matrix multiplication is an example.

A bilinear map can also be defined formodules. For that, see the articlepairing.

Let${\displaystyle V,W}$ and${\displaystyle X}$ be threevector spaces over the same basefield${\displaystyle F}$. A bilinear map is afunction$${\displaystyle B:V\times W\to X}$$such that for all${\displaystyle w\in W}$, the map${\displaystyle B_w}$$${\displaystyle v\mapsto B(v,w)}$$is alinear map from${\displaystyle V}$ to${\displaystyle X,}$ and for all${\displaystyle v\in V}$, the map${\displaystyle B_v}$$${\displaystyle w\mapsto B(v,w)}$$is a linear map from${\displaystyle W}$ to${\displaystyle X.}$ 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${\displaystyle B}$ satisfies the following properties.

• For any${\displaystyle \lambda \in F}$,${\displaystyle B(\lambda v,w)=B(v,\lambda w)=\lambda B(v,w).}$
• The map${\displaystyle B}$ is additive in both components: if${\displaystyle v_1,v_2\in V}$ and${\displaystyle w_1,w_2\in W,}$ then${\displaystyle B(v_1+v_2,w)=B(v_1,w)+B(v_2,w)}$ and${\displaystyle B(v,w_1+w_2)=B(v,w_1)+B(v,w_2).}$

If${\displaystyle V=W}$ and we haveB(v,w) =B(w,v) for all${\displaystyle v,w\in V,}$ then we say thatB issymmetric. IfX is the base fieldF, then the map is called abilinear form, which are well-studied (for example:scalar product,inner product, andquadratic form).

The definition works without any changes if instead of vector spaces over a fieldF, we usemodules over acommutative ringR. It generalizes ton-ary functions, where the proper term ismultilinear.

For non-commutative ringsR andS, a leftR-moduleM and a rightS-moduleN, a bilinear map is a mapB :M ×N →T withT an(R,S)-bimodule, and for which anyn inN,m ↦B(m,n) is anR-module homomorphism, and for anym inM,n ↦B(m,n) is anS-module homomorphism. This satisfies

B(r ⋅m,n) =r ⋅B(m,n)

B(m,n ⋅s) =B(m,n) ⋅s

for allm inM,n inN,r inR ands inS, as well asB beingadditive in each argument.

An immediate consequence of the definition is thatB(v,w) = 0_X wheneverv = 0_V orw = 0_W. This may be seen by writing thezero vector 0_V as0 ⋅ 0_V (and similarly for 0_W) and moving the scalar 0 "outside", in front ofB, by linearity.

The setL(V,W;X) of all bilinear maps is alinear subspace of the space (viz.vector space,module) of all maps fromV ×W intoX.

IfV,W,X arefinite-dimensional, then so isL(V,W;X). For${\displaystyle X=F,}$ that is, bilinear forms, the dimension of this space isdimV × dimW (while the spaceL(V ×W;F) oflinear forms is of dimensiondimV + dimW). To see this, choose abasis forV andW; then each bilinear map can be uniquely represented by the matrixB(e_i,f_j), and vice versa. Now, ifX is a space of higher dimension, we obviously havedimL(V,W;X) = dimV × dimW × dimX.

• Matrix multiplication is a bilinear mapM(m,n) × M(n,p) → M(m,p).
• If avector spaceV over thereal numbers${\displaystyle \mathbb{R}}$ carries aninner product, then the inner product is a bilinear map${\displaystyle V\times V\to \mathbb{R}.}$
• In general, for a vector spaceV over a fieldF, abilinear form onV is the same as a bilinear mapV ×V →F.
• IfV is a vector space withdual spaceV^∗, then thecanonical evaluation map,b(f,v) =f(v) is a bilinear map fromV^∗ ×V to the base field.
• LetV andW be vector spaces over the same base fieldF. Iff is a member ofV^∗ andg a member ofW^∗, thenb(v,w) =f(v)g(w) defines a bilinear mapV ×W →F.
• Thecross product in${\displaystyle {\mathbb{R}}^3}$ is a bilinear map${\displaystyle {\mathbb{R}}^3\times {\mathbb{R}}^3\to {\mathbb{R}}^3.}$
• Let${\displaystyle B:V\times W\to X}$ be a bilinear map, and${\displaystyle L:U\to W}$ be alinear map, then(v,u) ↦B(v,Lu) is a bilinear map onV ×U.

Suppose${\displaystyle X,Y,}$ and${\displaystyle Z}$ aretopological vector spaces and let${\displaystyle b:X\times Y\to Z}$ be a bilinear map. Thenb is said to beseparately continuous if the following two conditions hold:

1. for all${\displaystyle x\in X,}$ the map${\displaystyle Y\to Z}$ given by${\displaystyle y\mapsto b(x,y)}$ is continuous;
2. for all${\displaystyle y\in Y,}$ the map${\displaystyle X\to Z}$ given by${\displaystyle x\mapsto b(x,y)}$ is continuous.

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

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.

• IfX is aBaire space andY ismetrizable then every separately continuous bilinear map${\displaystyle b:X\times Y\to Z}$ is continuous.^[1]
• If${\displaystyle X,Y,\text{ and }Z}$ are thestrong duals ofFréchet spaces then every separately continuous bilinear map${\displaystyle b:X\times Y\to Z}$ is continuous.^[1]
• If a bilinear map is continuous at (0, 0) then it is continuous everywhere.^[2]

Let${\displaystyle X,Y,\text{ and }Z}$ belocally convexHausdorff spaces and let${\displaystyle C:L(X;Y)\times L(Y;Z)\to L(X;Z)}$ be the composition map defined by${\displaystyle C(u,v):=v\circ u.}$ In general, the bilinear map${\displaystyle C}$ 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:

1. give all three the topology of bounded convergence;
2. give all three thetopology of compact convergence;
3. give all three thetopology of pointwise convergence.

• If${\displaystyle E}$ is anequicontinuous subset of${\displaystyle L(Y;Z)}$ then the restriction${\displaystyle C{|}_{L(X;Y)\times E}:L(X;Y)\times E\to L(X;Z)}$ is continuous for all three topologies.^[1]
• If${\displaystyle Y}$ is abarreled space then for every sequence${\displaystyle {\left(u_i\right)}_{i=1}^{\mathrm{\infty }}}$ converging to${\displaystyle u}$ in${\displaystyle L(X;Y)}$ and every sequence${\displaystyle {\left(v_i\right)}_{i=1}^{\mathrm{\infty }}}$ converging to${\displaystyle v}$ in${\displaystyle L(Y;Z),}$ the sequence${\displaystyle {\left(v_i\circ u_i\right)}_{i=1}^{\mathrm{\infty }}}$ converges to${\displaystyle v\circ u}$ in${\displaystyle L(Y;Z).}$^[1]

• 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

• 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.

• "Bilinear mapping",Encyclopedia of Mathematics,EMS Press, 2001 [1994]

• Bilinear maps
• Multilinear algebra

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