That is, f∈V∗ and vf∈V are in a one-to-one correspondence. Given that an inner product is well defined in V, and since f,g∈V∗ uniquely corresponds to an element in V, the inner product on V∗ can be naturally defined as follows.
Definition
The inner product for the dual space V∗ of the Hilbert space (V,⟨⋅,⋅⟩V) is defined as below.
⟨f,g⟩V∗:=⟨vf,vg⟩V,f,g∈V∗(2)
Here, vf,vg∈V refers to the vector corresponding to f,g∈V∗ according to the Riesz Representation Theorem.
Explanation
The definition (2) can be re-expressed as follows due to (1).