This ticket provides more flexibility in the choice of symbols and indices for bases of free modules of finite rank and, consequently, for vector frames on manifolds.
In particular, the labels of the basis elements are no longer necessarily integers and can be customized, for instance:

sage: M = Manifold(3, 'M')
sage: e = M.vector_frame('e', indices=('x', 'y', 'z')); e
Vector frame (M, (e_x,e_y,e_z))

The symbol of each element of the basis can also be freely chosen, by providing a tuple of symbols instead of a single string; it is then mandatory to specify as well some symbols for the dual coframe:

sage: e = M.vector_frame(('a', 'b', 'c'), symbol_dual=('A', 'B', 'C')); e
Vector frame (M, (a,b,c))
sage: e[0]
Vector field a on the 3-dimensional differentiable manifold M
sage: e.coframe()
Coframe (M, (A,B,C))
sage: e.coframe()[0]
1-form A on the 3-dimensional differentiable manifold M

Besides, the slice operator is introduced for bases; it is now possible to write

sage: e[:]
(Vector field a on the 3-dimensional differentiable manifold M,
 Vector field b on the 3-dimensional differentiable manifold M,
 Vector field c on the 3-dimensional differentiable manifold M)

Internally, some refactoring of code has been performed. In particular, by renaming the FreeModuleCoBasis's attribute _form (the tuple containing the cobasis elements) to _vec, so that it matches the name of the FreeModuleBasis's attribute storing the basis elements, it has been possible to factor the method __getitem__ to the ABC Basis_abstract. Another improvement is getting rid of all methods _init_dual_basis in classes FreeModuleBasis, VectorFrame and CoordFrame thanks to the introduction of the class attribute _cobasis_class.

The functionalities introduced here provide a better user interface for #24623.

This work is part of the SageManifolds project, see #18528 for an overview.

CC: @tscrim

Component: geometry

Keywords: basis, vector frame, free moduie

Author: Eric Gourgoulhon

Branch/Commit: da06d12

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/24792