I'm struggling with exercise 6 from section 26 of Halmos' "Finite-dimensional Vector Spaces". The book asks
If $U$ and $V$ are finite-dimensional vector spaces, what is the dual space of $U' \otimes V'$?
Halmos defines $U \otimes V$ as the dual space of the space of bilinear forms on $U \oplus V$ (where $U \oplus V$ is the space of pairs of form $(x, y)$ with $x\in U$ and $y \in V$). Hence a literal interpretation of $U' \otimes V'$ is the double dual of the space of all bilinear forms on $U' \oplus V'$. But this interpretation does not take into account the fact that the spaces we are taking tensor product of (namely $U'$ and $V'$) are themselves dual of other spaces (namely $U$ and $V$). My question is is there a perhaps more natural interpretation of $U'\otimes V'$?