Tangent space of loop space. Let $\Omega$ be the path space of a riemannian manifold $M$. I have to define the tangent space of $\Omega$ in a path $\omega$, that I denote with $T_p \Omega$. I think that this space is the vector space of all vector fields $W$ defined along $\omega$ such that $W(0)=W(1)=0$. How can I define precisely $T_{\omega}\Omega$?
 A: Depending on which class of paths you consider, you will get different path spaces, which are smooth infinite-dimensional manifolds, modeled on different model spaces.
Let's assume you choose smooth paths. Then this is a manifold modeled on the Fréchet space $C^\infty_0(S^1, \mathbb{R}^n)$. Here the zero indicates that we mean the space of paths that are zero at a fixed point of $S^1$ (depending on your model of $S^1$).
Now for infinite-dimensional manifolds, you can define the (kinetic) tangent space exactly as in the finite-dimensional case, as equivalence classes of curves. Attention though: The algebraic tangent space (derivations) is much bigger and most likely not what you want.
In you example, there is a canonical isomorphism to the space you mentioned, the space $\Gamma^\infty_0(S^1, \gamma^*TM)$, of sections that vanish at your fixed point of $S^1$ ($\gamma$ is the point at which you take the tangent space).
It is a good exercise for getting used to these notions to work out this canonical isomorphism in this example.
