Reference for Taylor's Theorem $\mathbb R^n \to \mathbb R^m$

Question

I am looking for a reference/ book recommendation that in detail exhibits the theory behind the general Taylor theorem using jets for functions $f\colon \mathbb R^n \to \mathbb R^m$

\begin{align} f(x) = f(x_0) &+ Df(x_0)\cdot (x-x_0) + \frac 12 D^2f(x_0)\cdot (x-x_0)^{\otimes 2} + \ldots \\ &+ \frac {1}{k!}D^kf(x_0)\cdot (x-x_0)^{\otimes k} + R_{k+1}(x) \end{align}

Things that should be covered:

• Tensor products of Hilbert spaces
• Fréchet derivatives
• Jets
• The isomorphism $\underbrace{L(V,L(V,\ldots L(V,W))\ldots)}_{n \text{ times}} \cong L(V^{\otimes n}, W)$
• Some concrete example with $m,n\ge 2$, $k>2$

Answer 1

The best reference that I know of is Nonlinear Functional Analysis and its Applications Volume I: Fixed-Point Theorems by Eberhard Zeidler. Quick note, B-space means Banach space. Then, on page 148 and 149: