%%visits: 2 ## intuition The property sequilinearity ( a
generalisation of bilinearity) is implied. This is ⟨aŷ + bẑ, x̂⟩ = a*⟨ŷ, x̂⟩ + b*⟨ẑ, yhat⟩
## rigour inner_produc := A map is an inner_product, Where
V ∈ ℂn is
a finite vector. V × V → ℂ given by, $\left( \hat{x},\hat{y} \right)\to \left<
\hat{x,\hat{y}} \right>$ is an inner product if. - ⟨x̂, ŷ⟩* = ⟨ŷ, x̂⟩
- ⟨x̂, aŷ + bẑ⟩* = a⟨x̂, ŷ⟩ + b⟨x̂, ŷ⟩
- ⟨x̂, x̂⟩ ≥ 0 and
$\left< \hat{x},\hat{x}\right> = 0
\implies \hat{x} = \hat{0}$
tags :math: