Hyperhermitian inner product -- explicit examples requested

[Leditto]

Can anyone show me explicit examples of Hyperhermitian inner product?

 

[jvicens]

Sure, here are a few explicit examples of hyperhermitian inner products:

1. In the space of complex-valued continuous functions on a closed interval [a,b], we can define the inner product between two functions f and g as:

⟨f,g⟩ = ∫a^b f(x)g*(x) dx

where g*(x) is the complex conjugate of g(x). This inner product is hyperhermitian, meaning that ⟨f,g⟩ = ⟨g,f⟩*.

2. In the space of n-by-n complex matrices, we can define the inner product between two matrices A and B as:

⟨A,B⟩ = Tr(A†B)

where A† is the conjugate transpose of A. This inner product is hyperhermitian, as Tr(A†B) = Tr(B†A)†.

3. In the space of n-dimensional complex vectors, we can define the inner product between two vectors x and y as:

⟨x,y⟩ = x†y

where x† is the conjugate transpose of x. This inner product is also hyperhermitian, as x†y = (y†x)†.

These are just a few examples, but there are many other hyperhermitian inner products that can be defined in different spaces. Hope this helps!