Let $M$ be a finite-dimensional smooth manifold with atlas $\mathcal A$ and $p\in M$. There are various equivalent definitions of the tangent space $T_pM$:
- approach via derivations: the space of derivations of smooth functions: $C^\infty(M)\to \mathbb R$,
- approach via germs: the space of derivations of the germs at $p$, $C^\infty_p\to\mathbb R$,
- velocity vector approach: the set of curves $\sigma : (-\varepsilon, \varepsilon) \to M$ such that $\sigma(0)=p$, divided by an appropiate equivalence relation,
- physicists' approach: set of functions $X_p: \mathcal A\to \mathbb R^n$ that take a coordinate chart and give coordinates relative to this chart, assigning them according to the transformation rule.
- algebraic geometers' approach: the dual of $\mathfrak m_p/\mathfrak m_p^2$, where $\mathfrak m_p$ is the ideal of germs vanishing at $p$,
I wonder how one defines tangent space for different kinds of manifolds than smooth, especially:
- analytic/complex manifolds (i.e. one doesn't want to use partitions of unity as in 1.),
- $C^k$ manifolds for $1\le k<\infty$ (i.e. 5. is infinite dimensional),
- infinite-dimensional Banach manifolds.
Could you please provide references in which the above definitions are discussed? I would like to know which definitions are preferred (and to which are equivalent) in which contexts.