Cfrgtkky

In DoCarmo’s Riemannian Geometry book, it is written that if $phi$ is a smooth map from $M$ to $M$, $vin T_pM$, and f is a real-valued smooth map in a neighborhood of $phi(p)$, then we have $(dphi(v)f)phi(p)=v(fcirc phi)(p)$.

In this book, a tangent vector $v$ at $p$ is defined as a mapping of the set of real-valued smooth maps at p to $mathbb R$.

With those definitions and notations, the equality above does not make sense, since for example $(dphi(v)f)$ is already a real number, not a function, so $(dphi(v)f)phi(p)$ is wrong. So what does the author mean?

Appendix: this remark exactly in this form is used in proving $[X,Y](p)=lim_{trightarrow 0}1/ttimes(Y-dphi_t Y)(phi_t(p))$. Hence additionally I could not understand the left side of this equality.

You are correct. If $v in T_pM$, the expression $dphi(v) f$ is a real number.

But the expression "$(dphi(v)f)(phi(p))$" has a problem, even if $v$ is a vector field: What is "$dphi(v)$"? This is not necessarily well-defined as a vector field on $M$ if $phi$ is just smooth. Luckily it is in the case that $phi$ is a diffeomorphism. These things also make sense when $phi$ is a local diffeomorphism, but then you have vector fields defined on some open set $U$ and, accordingly, $phi(U)$. (In particular, it makes sense in the context of Lemma 5.5.)

As you mention, DoCarmo later uses the expression in the proof of another proposition:

(...) Accordingly,

$((dvarphi_tY)f)(varphi_t(p))=(Y(f circ varphi_t))(p)$ (...)

(For context, here $varphi_t$ is a local flow.) Therefore, as far as what he seems to be interested in is concerned, there is no problem in correcting his equality by saying that $v$ should be a vector field.

For more information, see the wikipedia section on pushforward of vector fields, for example.