Let $V$ and $W$ be vector spaces on a field $K$.

Question

Let $V$ and $W$ be vector spaces on a field $K$. Prove, constructing an explicit isomorphism, that $(V \oplus W) ^*$ is isomorphic to $V^* \oplus W^*$.

I need just one hint, please. How might one define an isomorphism?

Answer 1

Let $\varphi:V^{\ast}\oplus W^{\ast}\rightarrow(V\oplus W)^{\ast}$ be defined as $\varphi(v^{\ast},w^{\ast})(v+w)=v^{\ast}(v)+w^{\ast}(w)$, $\varphi$ is an isomorphism.

Let $\eta:(V\oplus W)^{\ast}\rightarrow V^{\ast}\oplus W^{\ast}$ be defined as $\eta(\omega^{\ast})_{1}(v)=\omega^{\ast}(v)$ and $\eta(\omega^{\ast})_{2}(w)=\omega^{\ast}(w)$ for $\omega^{\ast}\in(V\oplus W)^{\ast}$, then $\eta(\omega^{\ast})=(\eta(\omega^{\ast})_{1},\eta(\omega^{\ast})_{2})\in V^{\ast}\oplus W^{\ast}$ and $\eta$ is an isomorphism.

Answer 2

Hint: take an element of $V^*\oplus W^*$, and think how you could make it a functional acting on $V\oplus W$.

Or, taking the opposite approach, take a functional from $(V\oplus W)^*$, and think how you can obtain a functional on $V$ and another one on $W$.