bilinear map

Let $R$ be a ring, and let $M$ , $N$ and $P$ be modules over $R$ . A function $f\colon M\times N\to P$ is said to be a bilinear map if for each $b\in M$ the function $h\colon N\to P$ defined by $h(y)=f(b,y)$ for all $y\in N$ is linear (http://planetmath.org/LinearTransformation) (that is, an $R$ -module homomorphism), and for each $c\in N$ the function $g\colon M\to P$ defined by $g(x)=f(x,c)$ for all $x\in M$ is linear. Sometimes we may say that the function is $R$ -bilinear, .

A common case is a bilinear map $V\times V\to V$ , where $V$ is a vector space over a field $K$ ; the vector space with this operation then forms an algebra over $K$ .

If $R$ is a commutative ring, then every $R$ -bilinear map $M\times N\to P$ corresponds in a natural way to a linear map $M\otimes N\to P$ , where $M\otimes N$ is the tensor product of $M$ and $N$ (over $R$ ).

Title	bilinear map
Canonical name	BilinearMap
Date of creation	2013-03-22 15:35:47
Last modified on	2013-03-22 15:35:47
Owner	yark (2760)
Last modified by	yark (2760)
Numerical id	11
Author	yark (2760)
Entry type	Definition
Classification	msc 13C99
Synonym	bilinear function
Synonym	bilinear operation
Synonym	bilinear mapping
Synonym	bilinear operator
Synonym	bilinear pairing
Synonym	pairing
Related topic	Multilinear
Related topic	BilinearForm
Related topic	ScalarMap
Defines	bilinear