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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 $T_pM$:

  1. approach via derivations: the space of derivations of smooth functions: $C^\infty(M)\to \mathbb R$,
  2. approach via germs: the space of derivations of the germs at $p$, $C^\infty_p\to\mathbb R$,
  3. velocity vector approach: the set of curves $\sigma : (-\varepsilon, \varepsilon) \to M$ such that $\sigma(0)=p$, divided by an appropiate equivalence relation,
  4. 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.
  5. 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.

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  • $\begingroup$ John Lee's Introduction to Smooth Manifolds does all the approaches except (as far as I've seen) not 5. $\endgroup$ – Chris Mar 5 at 2:42
  • $\begingroup$ As far as i remember Lee's smooth manifold only did for algebraic approach and the just mention few others. However, i found a decent explanation of all approach in Jeff Lee's manifolds $\endgroup$ – Sou Mar 5 at 4:17
  • $\begingroup$ @Sou I agree - John Lee's book covers the algebraic approach and mentions others. It also mentions that the definition with germs works in the analytic/complex case. I'm looking for something explaining how these definitions tie together (or become irrelevant) in other categories than just $C^\infty$ manifolds. Unfortunately Jeff Lee's book doesn't cover other categories either... $\endgroup$ – Paweł Czyż Mar 5 at 19:41
  • $\begingroup$ I didn't know much about that but probably Lang's book or KMS may help. $\endgroup$ – Sou Mar 6 at 7:41

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