The following is an excerpt from Chern's Lectures on Differential Geometry:
I don't see how the proof shows the other direction of the set inclusion. Would anybody explain the logic in the "furthermore" part of the proof?
The following are definitions of the notations:
This is essentially from A.Sh's and Ted Shifrin's comments.
"a symmetric tensor is invariant under the symmetrizing mapping" shows that $S_r$ is an identity map on $P^r(V)$ and thus $P^r(V)=S_r(P^r(V))\subset S_r(T^r(V))$.
Similarly, "an alternating tensor is invariant under the alternating mapping" shows that $\Lambda^r(V)=A_r(\Lambda^r(V))\subset A_r(T^r(V))$.
What confused me was the word "invariant". It is not only that $P^r(V)$ is invariant under $S_r$, which would give $P^r(V)\supset S_r(P^r(V))$, but also that every element in $P^r(V)$ is invariant under $S_r$, which means that $S_r$ is an identity map on $P^r(V)$ and thus $P^r(V)=S_r(P^r(V))$.
(Yes, this is a stupid question.)