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Let $A$ a open set in a Hilbert Space $H$, suppose that $f:A\to F$ is differentiable at $a\in A$ and that $||f(x)||=c$ forall $x\in A$. Show that range of $Df(a)$ is contained in the subspace $\{f(a)\}^{\perp}$.

Note: $f$ is differentiable in $a$ iff exist $Df(a)\in \mathcal{L}(E, F)$ so that $$\lim_{h\to 0}\frac{||f(a+h)-f(a)-Df(a)h||_F}{||h||_E}=0$$

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  • $\begingroup$ What is $F$. Is it $H$? $\endgroup$
    – Tomás
    Jul 5 '13 at 16:38
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By definition $$\{f(a)\}^\perp=\{x\in F:\ (x,f(a))=0\}$$

By hypothesis you have that $(f(x),f(x))=c$, where $c$ is come constant. If $f$ is differentiable in $a$, we can derivate the last equality to conclude that $$(f'(a)y,f(a))=0, \forall\ y\in H$$

The last equality implies that $f'(a)y\in \{f(a)\}^\perp$ for all $y\in H$.

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