My question is about the tangent space of a manifold in some given point.
Let $M$ be a differential manifold and $(U,\varphi)$ a chart around a given point $p$ of $M$ .
My question is : Is that the tangent space to $U$ in $p$ is equal to the tangent space to $M$ in $p$ ?
$i.e$ $$T_{p}U = T_{p}M$$ Can anyone help me please ? Thank you in advance.
It is essentially the same, and is usually taken to be exactly equivalent. In reality, if $U\subset M$ and $p\in U$, then $T_pU$ and $T_pM$ are isomorphic. In fact, $d\iota_p:T_pU\to T_pM$ is the required isomorphism, for $\iota:U\to M$ the inclusion. The proof is easy enough, and is also found in Lee's Smooth Manifolds book (prop 3.9).
Edit: If relevant, he uses the definition of $T_pM$ as the space of derivations at $p$. If you're using a different definition (equivalence classes of curves/their tangents), then the proof is slightly different, but still works about the same as you'd expect.