If $V$ is a space with inner product ($\cdot,\cdot$). If $f:V\rightarrow \mathbb{R}$, $$f(u)=(u,u)$$ find Frechet derivative $f'(u)$ Can anybody help me? Thanks
2025-04-02 09:21:57.1743585717
frechet derivative in space with inner product
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if we set $f'(u)=(2u,1)$ then $\frac{|| 2(u,h) +(h,h) - (2u,1)h||}{\sqrt{|(h,h)|}} =\frac{|| 2(u,h) +(h,h) - (2u,h)||}{\sqrt{|(h,h)|}} = \frac{||(h,h)||}{\sqrt{|(h,h)|}} = \frac{|(h,h)|}{\sqrt{|(h,h)|}}=\sqrt{|(h,h)|}$ wich goes to $0$ as $h \rightarrow 0$