TITLE: Is there a name for this map induced by bilinear forms?
QUESTION [6 upvotes]: Let $V$ be a real vector space. A bilinear form $\langle \rangle:V\times V\to {\mathbb{R}}$ induces a linear functional $\theta$ on the tensor product $V\otimes V$ given by sending the finite sum $\sum_i v_i\otimes w_i $ to $\sum_i \langle v_i,w_i\rangle$.
Is there a name for this induced linear functional?
In addition, if the bilinear form is symmetric, then this linear functional $\theta$ respects the natural involution on $V\otimes V$. That is $\theta(v\otimes w)=\theta(w\otimes v)$.

REPLY [2 votes]: A bilinear form on $V$ (if non-degenerate) lets you identify $V$ with $V^\star$:
$v \mapsto \langle v, \cdot \rangle$
In this case, your map is just a contraction of the identity map on $V^\star \otimes V$ (which, considering our identification, is the same as the identity on $V \otimes V$).
http://en.wikipedia.org/wiki/Tensor_contraction