# Tensor Products, various defintions

I came across a definition for the tensor product which differs from the standard definition. This book defined the tensor product of vector spaces $V$ and $W$ as the space $L(V,W,\Bbb K)$ of bilinear maps into the field $\Bbb K$. Is this definition of tensor product only equivalent to the standard definition in the finite dimensional case?

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it would help if you included the precise definition you encountered and the definition you refer to as the standard one. –  Ittay Weiss Mar 23 '13 at 23:23
I've definitely seen this definition of tensor product used for Hilbert spaces, for instance (if memory serves me) in Volume 1 of Kadison--Ringrose's textbook on operator algebras, with the proviso that you consider only continuous (in the appropriate sense) bilinear forms. –  Branimir Ćaćić Mar 23 '13 at 23:39
@Mark The definition you quote is only suitable for finite-dimensional vector spaces. It actually defines $(V \otimes W)^*$. –  Zhen Lin Mar 24 '13 at 0:11
is it $L(V\times W,{\Bbb{K}})$ in the OP?? –  janmarqz Jan 11 at 18:42