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While reading a text in multivariable calculus , a following definition of higher order derivatives in normed linear spaces is given like this :

If $U\subset E$ is open in some normed linear space $E$ and $f:U\to F$ is a function such that $f''$ exists at each $a\in U$ ($F$ is NLS as well) , then $a\mapsto f''(a)$ belongs to $\mathcal{L}(E,\mathcal{L}(E,F))$ and that space is isomorphic to $\mathcal{L}_2(E,F)$ , i.e. the space of all continuous symmetric bilinear forms from $E\times E$ to $F$ .

What I don't understand is how to explicitly construct this isomorphism . Any help is appreciated in this regard .

am_11235...
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  • Connected : https://math.stackexchange.com/q/189568/305862 – Jean Marie May 23 '22 at 17:34
  • A different point of view would be to consider the tensor product : linear functions over $E \otimes E$ is simpler than bilinear functions on $E \times E$... – Jean Marie May 23 '22 at 17:35

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