On dual spaces and inner products

Question

Let V be a vector space over $\mathbb{C}$ equipped with an inner product $\langle\, , \rangle:V\times V\mapsto\mathbb{C}$. I need to prove that any linear function $\phi:V\mapsto\mathbb{C}$ (element of the dual space) can be consider as the inner product of a vector $\vec{v}$ with the rest of the space. That is, for every $\phi \in V^*$, there exists a vector $\vec{v}\in V$ such that for every $\vec{u}\in V$ we have $\phi(\vec{u})=\langle\vec{v},\vec{u}\rangle$. I don't even know where to start! Any help s appreciated.

Answer 1

Hint: Consider the map $V\to V^\ast$ sending $\vec{v}$ to $\langle\vec{v},-\rangle$. Show that it is injective, and conclude that it is surjective.

Alternatively, you could also try to explicitly construct such a vector. To start, choose an orthonormal basis.

Note: This is false when $V$ is infinite-dimensional!