Question about perfect pairings

Solution 1:

I'll phrase this for general fields $k$. Let $B : V \times W \to k$ be a $k$-bilinear map. (By the universal property of tensor products, this is the same thing as a $k$-linear map $V \otimes W \to k$.) Then, $B$ induces two $k$-linear maps $L : V \to W^*$, $R : W \to V^*$ by partial evaluation: $$L(v) = (w \mapsto B(v, w))$$ $$R(w) = (v \mapsto B(v, w))$$ We say $B$ is a perfect pairing if both $L$ and $R$ are isomorphisms. If $V$ and $W$ are both finite-dimensional, then this is equivalent to non-degeneracy of $B$. This means it suffices to check that

• For each $v$, if $B(v, w) = 0$ for all $w$, then $v = 0$.
• For each $w$, if $B(v, w) = 0$ for all $v$, then $w = 0$.