Alternative proof of Taylor's formula by only using the linear approximation property So a function $f: E \to F$ between the normed spaces $E,F$ is called differentiable in $x \in E$ if there exists a bounded linear map $Df(x): E \to F$ such that for every $h \in E$ we have $$f(x+h)=f(x)+Df(x)h + o(||h||). \tag{1}$$
If $f$ is differentiable for every $x \in E$ and $Df: x \mapsto Df(x)$ is differentiable for every $x \in E$ too we get analogously
$$Df(x+e)=Df(x)+D^2f(x)e+o(||e||). \tag{2}$$
Then $f$ is called twice differentiable and for every $h\in E$ we have the &quot;Taylor expansion of second degree&quot;
$$f(x+h)=f(x)+Df(x)h+\frac{1}{2}D^2f(x)[h] + o(||h||^2), \tag{3}$$
where $D^2f(x)[h]:=(D^2f(x)h)h$ for better readability.
I have two questions:

How can $(3)$ be proven without resorting to the &quot;standard proof&quot; of using integrals? I want to show it by only using the linear approximations  given in $(1)$ and $(2)$. Inserting $(2)$ in $(1)$ doesn't result in something useful though. Can this be done?
Can $(3)$  be used as an alternative definition off twice-differentiability? Analogously what about  the general case of $n$-times differentiability:
$$ f(x+h) = f(x) + \sum_{j=1}^{n} \frac{1}{j!} D^jf(x)[h] + o(\|h\|^n) \tag{4}$$
A few days back I wrote an answer with some detail about Taylor polynomials for maps between Banach spaces. You can see my answer here. What I proved is that (I'm sorry about the differences in notation)


  Taylor Expansion Theorem:
  
  Let $V$ and $W$ be Banach spaces over the field $\Bbb{R}$, let $U$ be an open subset of $V$, and fix a point $a \in U$. Let $f:U \to W$ be a given function which is $n$ times differentiable at $a$ (in the Frechet differentiable sense). Define the Taylor polynomial $T_{n,f}:V \to W$ by
  \begin{equation}
T_{n,f}(h) = f(a) + \dfrac{df_a(h)}{1!} + \dfrac{d^2f_a(h)^2}{2!} + \dots + \dfrac{d^nf_a(h)^n}{n!}
\end{equation}
  Then, $f(a+h) - T_{n,f}(h) = o(\lVert h \rVert^n)$. 
  
  Explicitly, the claim is that for every $\varepsilon &gt; 0$, there is a $\delta &gt; 0$ such that for all $h \in V$, if $\lVert h \rVert &lt; \delta$, then
  \begin{equation}
\lVert f(a+h) - T_{n,f}(h) \rVert \leq \varepsilon \lVert h \rVert^{n}.
\end{equation}


The proof is pretty short if you know what you're doing.
The idea is to use induction, and most importantly, the mean-value inequality for maps between Banach spaces. I don't think it is possible to derive $(3)$ directly from $(1)$ and $(2)$ alone, because $(2)$ talks about how the derivative $Df$ changes, while $(1)$ talks about how the function $f$ changes, and ultimately $(3)$ talks about how much $f$ changes. So, you somehow have to relate changes in $Df$ to changes in $f$... this is roughly speaking, what the mean-value inequality does.

The proof I showed in my other answer is pretty much from Henri Cartan's excellent book Differential Calculus. Also, Henri Cartan's book has a proof of the mean-value inequality which doesn't rely on integrals. Alternatively, you can take a look at Loomis and Sternberg's book Advanced Calculus. Here, they prove the mean value inequality in a rather elementary way without integrals, and it's also a relatively short proof. It is proven in Chapter 3, Theorem 7.4 (which uses theorem 7.3); this is on page 148-149 of the book (I prefer this proof to Cartan's proof). 

For your other question, I assume you mean the following:


  Does the existence of a polynomial $P$, which equals $f$ up to order $2$ at $x$ imply that $f$ is twice differentiable at $x$? Or more precisely, does the existence of a continuous linear map $A_1:E \to F$, and a symmetric continuous bilinear map $A_2:E \times E \to F$ such that for all $h \in E$,
  \begin{equation}
f(x+h) = f(x) + A_1(h) + A_2(h,h) + o(\lVert h \rVert^2)
\end{equation}
  imply that $f$ is twice Frechet differentiable at $x$?


The answer to this question is no. We can see this even in the single variable case (this following example is from Spivak's Calculus, page 413, 3rd edition). Take $E=F=\Bbb{R}$, and define $f: \Bbb{R} \to \Bbb{R}$ by
\begin{equation}
f(x) = 
\begin{cases}
x^{n+1}&amp; \text{if $x$ irrational} \\
0 &amp; \text{if $x$ rational}
\end{cases}
\end{equation}
($n\geq 2$). Then choose $x=0$ and the zero polynomial $P \equiv 0$. It is easy to verify that
\begin{equation}
f(0 + h) = 0 + o(|h|^n)
\end{equation}
However, if $a \neq 0$, then $f'(a)$ doesn't exist so $f''(0)$ is not even defined. Hence, what this shows is that the existence of a well-approximating polynomial does not guarantee that the function is sufficiently differentable.
The differential Df(x) is a linear map that approximates your function f, according to your relation (1).  And that linear map depends on x (the point around which you consider the Taylor expansion).  The map Df , which associates a vector to a linear map doesn't have to be linear indeed.  The second differential  DDf(x)   is a linear map that associates a vector to a linear map , and so on. Your relation (1) tells us that the differential of f exists (definition). Your relation (2) tells us that the differential of Df exists (definition). Your relation (3) tells us that there is a valid finite second order (degree) Taylor expansion of f at x (also a definition). From (1) and (2) you cannot derive (3). That's not always possible  even for real function , even more so in this abstract setting.  In a valid, finite Taylor expansion,  the remainder must grow slower than the last term of the Taylor expansion.  And no, (3) is not equivalent to second order differentiability of f.