I am tryig to sort of how inner products and duals are related. According to https://en.wikipedia.org/wiki/Dual_pair it donst seem possible to use an inner product when we define the natural dual pairing, yet Riesz theorem tells us that any functional is given by the innerproduct.
Does anyone have an idea how this add up?
From Fichers comment,
No, $⟨⋅,⋅⟩$ is the pairing between the inner product space and its dual. $⟨⋅∣⋅⟩$ is the inner product. And $⟨⋅∣w⟩$ is the image of ww under the Riesz map, that is, the linear functional $φ_{w}:v↦⟨v∣w⟩$. This way, we can write it as $⟨v,φ_{w}⟩=⟨v∣w⟩$