STATEMENT: Suppose V is a finite dimensional vector space with an inner product $\langle *,*\rangle$. And $\phi :V\rightarrow V^*$ is the isomorphism $\phi(v)=\langle v,*\rangle$ The inner product, together with the isomorphism $\phi$, define an inner product on $V^*$.
QUESTION: What is the particular inner product that is induced on $V^*$? Are they talking about the inner product $\langle u,v\rangle_V=\langle\phi(u),\phi(v)\rangle_{V^*}$?