I was trying to understand the following sentence in some notes I am reading:
Let $M$ be a manifold with boundary.
At any point $p \in {\partial}M$ there is a canonical subspace $T_{p}({\partial}M) \subset T_{p}(M)$; the quotient space is the a real line $\nu_{p}$.
I know of $T_{p}(M)$ as the vector space consisting of operators or derivations $\nu: F(M) \rightarrow \mathbb{R}$ where $F(M)$ is the algebra of smooth functions from $M \rightarrow \mathbb{R}$.
Does this natural subspace involve some sort of imbedding of a $F({\partial}M)$ into $F(M)$?
I apologize if this question is obvious.
No, it involves projecting $F(M)\to F(\partial M)$, i.e. observing that a smooth function on $M$ restricts to a smooth function on $\partial M$. This seems to go the wrong directions, but now the derivations come into play: a derivation $ F(\partial M)\to\mathbb R$ gives rise to a map $F(M)\to F(\partial M)\to\mathbb R$ (that is also a derivation, as you may check).