I am interested in learning what terminology and notation, other than the small-$o$ notation, exist in the literature for describing the maps $\mathbf{R}\to\mathbf{R}$ that vanish at $0$"faster than linearly." Using the small-$o$ notation, they are the maps of the class $o(x)$ as $x\to 0$.
The terminology I am looking for should also be applicable to maps $\mathbf{R}^m\to\mathbf{R}^n$, and preferably to maps between any topological vector spaces.
Usually, in the case of such maps between normed vector spaces, they are described as $o(\Vert\boldsymbol{v}\Vert)$ as $\boldsymbol{v}\to\mathbf{0}$. However, the choice of the norm here is irrelevant, as long as the topology is defined.
I find that this kind of maps plays a fundamental role in basic analysis and in basic differential geometry, so describing them as $o(\Vert\boldsymbol{v}\Vert)$ seems awkward, especially because in the finite-dimensional case the choice of the norm is completely irrelevant.
By the way, I suspect that the definition of this class of maps can be generalized to be meaningful for maps between any topological groups.
This class of maps is closely related to an equivalence relation on maps between affine spaces (or differentiable manifolds) at a given point, according to which two maps are equivalent if at the given point their values coincide and the difference of the associated linear maps between the tangent spaces is vanishing at $0$ faster than linearly.I've seen a few terms for this equivalence relation.Such maps of affine spaces may be said
- to be tangent to each other at the given point (Deudonné), or
- to have contact of order $\ge 1$ at the given point (Bourbaki).