(follow-up from #30092)

We define a category InnerProductSpaces with

• parent method dual (which returns self),
• element method inner_product.

A complication for complex vector spaces: FreeModule defines:

• dot_product - which ignores inner_product_matrix and does not conjugate
• hermitian_inner_product - which ignores inner_product_matrix but conjugates the left factor (self)
• inner_product - which uses inner_product_matrix but does not conjugate.

Perhaps we need to add an extra parameter inner_product_hermitian to FreeModule. For a complex base ring, the module would only be added to the category of InnerProductSpaces if this is True.

MatrixSpace elements currently do not have inner_product.

References:

• Hermitian inner product #10683 ... where hermitian_inner_product was introduced
• numpy.inner and numpy.dot, numpy.tensordot, numpy.einsum do not conjugate.
• numpy.vdot conjugates the left factor; and also handles the inner product of matrices in the same way.
• SymPy's Matrix.dot has parameters hermitian and conjugate_convention that control if and how factors are conjugated.

See also: https://en.wikipedia.org/wiki/Pseudo-Euclidean_space

CC: @tscrim @orlitzky @egourgoulhon @mjungmath @kcrisman

Component: linear algebra

Issue created by migration from https://trac.sagemath.org/ticket/30218