I was thinking about the derivative, and I wanted to make sure I’m thinking about it the right way. Suppose we have a $C^{\infty}$ function $f: {V}\to \mathbb{R}$, where $V$ is a finite-dimensional real inner product space. The Taylor expansion up to first term is
$$ f(x+h) = f(x) + \langle \nabla f(x), h \rangle + o(\|h\|). $$
So the first derivative, $\nabla f(x)$, is a function on $V$ taking values in $V^*$, it seems to me (we don’t really need the inner product here). Up to two terms, the Taylor expansion is
$$ f(x+h) = f(x) + \langle \nabla f(x), h \rangle + \langle \nabla^2 f(x) h, h \rangle + o(\|h\|^2). $$
So the second derivative, $\nabla^2 f(x)$, is a function taking values in $\mathcal{L}(V, V^*) = (V^{*})^{\otimes 2}=(V^{\otimes 2})^*$. It so happens that in this case, we want to plug in $h \otimes h$. Question 1: Would there be a related situation in which we would have different things than $h \otimes h$ plugged in? How about tensors of rank $>1$?
The third derivative should be a function taking values in $\mathcal{L}(V, (V^{*})^{\otimes 2})= (V^{*})^{\otimes 3} = (V^{\otimes 3})^*$. Again we would plug in $h \otimes h \otimes h$.
Then if we had a $C^\infty$ function $g: V \to W$, the derivative would take values in $\mathcal{L}(V,W)=V^* \otimes W$. The second derivative would take values in $\mathcal{L}(V, V^* \otimes W)=(V^*)^{\otimes 2}\otimes W$. Question 2: Am I right that there is a Taylor expansion here like
$$ g(y + k) = g(y) + Dg(y)k + D^2g(y)(k \otimes k) + o(\|k\|^2)? $$
Question 3: How can we describe the restriction of a function $\mathcal{L}(V^{\otimes 3}, W)$ to tensors of the form $h \otimes h \otimes h$?
Any related articles or resources would be appreciated. Thanks!