Question

Let $X$ be a Frechet or Banach manifold. We can define tangent vectors by equivalence classes of smooth curves. But, we could also define them as derivations of germs of smooth functions. Do these two notions agree in this infinite dimensional setting?

Answer 1

I believe the condition for these two definition to agree is that the Frechet manifold be nuclear. See section 28 (proposition 28.7) of "The convenient setting of global analysis" by A. Kriegl and P.W. Michor.

Answer 2

To expand on Gjergji's answer a little:

According to Kriegl and Michor, the key for the identification is two properties on the model space:

1. It be reflexive
2. It have the bornological approximation property

Reflexivity is needed because, essentially, of what André mentions: linear maps in to a space are not always the same as linear maps out of linear maps out of the space! The derivational definition of tangent vectors is as operators on cotangent vectors (even if dressed up a derivations on germs) so it's a "dual of a dual" and you need reflexivity to get back where you started.

Approximation properties (of which there are many) are a way of saying that linear maps from one space to another aren't too messy, by which we mean that they behave a little like matrices. Matrices themselves correspond to things in $E \otimes E'$, but as we're in infinite dimensions then that's only finite rank operators. Still, as we're Grown Ups, we can handle approximations and so it's enough for us that any linear transformation can be approximated by a matrix. With the strong (or operator) topology, this only gets us compact operators, but with a weak topology we have a chance of getting the whole thing. The different types of approximation property correspond to different topologies on linear maps.

There are spaces (even Banach spaces) that don't have the requisite approximation property, but they're "not nice" spaces and also not ones that are usually encountered. All the "most common" spaces have the approximation property.

In particular, nuclear Frechet spaces are reflexive and have the approximation property, so - as Gjerji says - that's sufficient. However, Hilbert spaces are also reflexive and have the approximation property.

The last thing I have to say on this is that if your real question is "What's the right notion of tangent vector in infinite dimensions?" then you should look at the chart in Kriegl and Michor's book in section 33.21 (p351 in the current version online). From that chart, it's clear that the "right" definition is as equivalence classes of curves.

Answer 3

Let $X$ be a Banach or Fréchet vector space.
Then the following two notions already don't agree:
1) linear map $\mathbb R\to X$
2) linear functional on the space of linear functions.