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Consider the space $C\bigl([a,b];\mathbb{R}\bigr)$ equipped with the $\sup$ norm. Define the operator $\mathfrak{f} : C\bigl([a,b];\mathbb{R}\bigr) \to C\bigl([a,b];\mathbb{R}\bigr) \: \text{by} \ \ \mathfrak{f}(\varphi)(t) = \int_{a}^{b}(\varphi(s))^{3} ds \cdot \varphi(t), \ \ \text{for} \ \varphi\in C\bigl([a,b];\mathbb{R}\bigr)$

  • Now for a given $\chi\in C\bigl([a,b];\mathbb{R}\bigr)$,I want to find a linear operator $\mathscr{L} : C\bigl([a,b];\mathbb{R}\bigr) \to C\bigl([a,b];\mathbb{R}\bigr)$ satisfying $\lim_{||\varphi||_{\infty}\to 0} \: \frac{\mathfrak{f}(\chi+\psi)-\mathfrak{f}(\chi)-\mathscr{L}\varphi}{||\varphi||_{\infty}}=0.$

  • I also want to show $\mathscr{L}$ is continuous. I know that it suffices to show $\mathscr{L}$ is bounded.

  • Also I want to calculate the derivative of $D\mathfrak{f}(\chi)$ of $\mathfrak{f}$ at $\chi$ $?$

A solution would be of great help.

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    There is a typo in the difference quotient. It should be $\varphi$ instead of $\psi$ (I guess!).2012-12-16

1 Answers 1

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You can proceed as follows: Consider the difference quotient $\frac{\mathfrak{f}(\chi + \epsilon\phi) - \mathfrak{f}(\chi)}{\epsilon}$ and then pass to the limit $\epsilon\to 0$. You should observe that the result will be linear in $\phi$ and call the corresponding linear operator $\mathcal{L}$. In your case, Davide already told you what to do (this goes through without complications).

For your last bullet: What you did was calculating the Gateaux-derivative of $\mathfrak{f}$ (there are several other notions of derivatives around, so you should be more precise in what kind of derivative you want).

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    Could you please elaborate more.2012-12-15