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If $ f:R \to H$ is the map such that $H$ is a Hilbert Space and $\langle f(x), f(x)\rangle =1$.

My problem involves differentiating the equation involving the inner product above to get a relationship between $f(x)$ and $f'(x)$.

Kindly help!! Thanks & regards.

mrtaurho
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Devendra Singh Rana
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  • $<f',f>+<f,f'> = 0$ --> $<f,f'> = 0$ – mathworker21 Aug 14 '19 at 13:14
  • What have you used @mathworker21 ? – Devendra Singh Rana Aug 14 '19 at 13:16
  • https://math.stackexchange.com/questions/3155664/product-rule-for-vector-output-functions/3244485 this answer of mine might be helpful for you. it's basically a generalisation of the product rule (the inner product is a bounded bilinear map so what I called $\omega$ in my answer can be replaced by the inner product) – peek-a-boo Aug 14 '19 at 13:21
  • Nice, Can you suggest any readings for the proof of the same? @peek-a-boo – Devendra Singh Rana Aug 14 '19 at 13:28
  • proof of the "same"? do you mean product rule? if yes, then take a look at the book by Loomis and Sternberg which I referenced in that answer – peek-a-boo Aug 14 '19 at 13:30
  • Can I use chain rule, to find the desired Derivative as I know the Derivative of a bilenear form but for the arguments being independent variables??? @peek-a-boo – Devendra Singh Rana Aug 14 '19 at 13:38
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    yes that's how this "product rule" is proven in the book I referenced. (pretty much any differentiation rule can be proven from the chain rule; for example sum quotient, product rules can all be derived from chain rule) – peek-a-boo Aug 14 '19 at 13:41

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