I define tangent space T to differentiable manifold, in point p, via equivalence class of curves. The condition for this equivalence is $(\varphi\circ\gamma_1)'(0)=(\varphi\circ\gamma_2)'(0)$ for some map $(U,\varphi)$. Another characterization of this condition is $(f\circ\gamma_1)'(0)=(f\circ\gamma_2)'(0)$ for every real valued funcion from $C^k(M)$.
Usually, vector space structure of T is define in some map U, p $\in U$. My question is follows. Can I define vector space structure of T without map? I mean, in terms of real valued functions?
Let $\gamma$ be curve, $t\mapsto\gamma(t)$. If $[\gamma]$ is class from T, then I can define scalar multiplication $c[\gamma]$, like $c\gamma:t\mapsto\gamma(ct)$. But I do not know how can I define addition.