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Now we know that given two Banach spaces $E$ and $F$ and a function $\ f:E \to F $ , the derivative $ Df(x) $ is a linear map from $E$ to $F$ at some point $ x $ in $E$. Briefly $ \ Df: E \to L(E,F) $ where $ L(E,F) $ is space of all linear mappings from $E$ to $F$. Now my question is how would a second order derivative look like. Since the derivative map of a linear map is linear map itself does this mean $ \ D^2(f(x))=D(f(x)) $ which doesn't seem likely to me. Or is it something else? Also, is the following right? $$D^2(f): E \to L(L(E,F),F) $$

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A related question: math.stackexchange.com/q/123007/8157 –  Giuseppe Negro Sep 1 '12 at 8:15
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3 Answers

up vote 2 down vote accepted

Let $U$ be an open set of $E$ and $f : U \longrightarrow F$. The differential of $f$ is a map:

$$ D: U \longrightarrow L(E,F), \, \, \ u\mapsto Df(u) $$

And we can define,

$$ D^r = D(D^{r-1})(f) : U \longrightarrow L(E,L^{r-1}(E,F)) $$

Since $$ L(E,L^{r-1}(E,F)) \simeq L^r(E,F) $$ we usually use $$D^r : U \longrightarrow L^r(E,F) $$

Where $L^r(E,F) = L(E \times \cdots \times E,F)$ ($r$-times).

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Why not just write $L(E^r,F)$ instead of $L^r(E,F)$? –  Ben Millwood Sep 1 '12 at 9:28
    
Because it's the commonly used notation. –  Ilies Zidane Sep 1 '12 at 9:38
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Just as the first derivative is the linear term in the Taylor expansion: \begin{equation} f(x+h)=f(x)+Df(x)h+\text{higher order}, \end{equation} the second derivative should be the quadratic term in the Taylor expansion: \begin{align} D^2f(x)\ \text{should be}\ Q\colon E \times E \to F, \text{where}& f(x+h)=f(x)+Df(x)h+\frac{1}{2}Q(h, h)+\ldots \end{align} Here $Q$ is linear in each variable, that is, $Q$ is a bilinear operator.

Since $Df$ is a mapping from $E$ to the Banach space $L(E;F)$ it can be differentiated yielding a mapping from $E$ to $L(E; L(E;F))$, as you point out in the first post. But, as Ilies notes, this last space is exactly the space of bilinear operators, up to the natural identification \begin{equation} Q(h,k)=Q(h)(k), \end{equation} and, defining $D^2f(x)$ to be the derivative of $Df(x)$ with this identification, turns out that Taylor's formula holds: \begin{equation} f(x+h)=f(x)+Df(x)h+\frac{1}{2}D^2f(x)(h, h)+o\left(\lVert h \rVert^2\right). \end{equation} This completely justifies our definition.

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Thank you, that was very clear and precise. –  Vishesh Sep 1 '12 at 9:06
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Regarding Ilies Zidane's answer, it's important to point out that it isn't just matter of widely used notation, but actually $\mathcal{L}(E^r,F)$ and $\mathcal{L}^r(E,F)$ are just not the same thing: the former is the space of all linear continuous maps from $E^r$ to $F$, while the latter is the one of all $r$-linear continuous maps from $E$ to $F$. So, for instance, if $E=F=\mathbf{R}$, then $\mathcal{L}(\mathbf{R}^r,\mathbf{R})={(\mathbf{R}^r)}^*$ is an $r$-dimensional vector space, while $\mathcal{L}^r(\mathbf{R},\mathbf{R})$ is just a $1$-dimensional one!

As for $\mathcal{L}^r(E,F)$ and $\mathcal{L}(E,\mathcal{L}^{r-1}(E,F))$, they aren't equal to each other either, but isomorphic as Banach spaces.

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