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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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First, we should denote $(\mathfrak f(\varphi))(t)$, since $\varphi(t)$ is a real number, and $\mathfrak f$ acts on functions, not numbers. I think you mean "$\mathcal L$ is continuous" not $\chi$. –  Davide Giraudo Dec 15 '12 at 16:52
    
@DavideGiraudo Thanks, is it ok now. –  Mithun Dec 15 '12 at 16:56
    
There still is a typo in the second item in the list ($\mathcal L$ instead of $\chi$), and the map $\mathfrak f$ is not linear (so the title needs an edit). –  Davide Giraudo Dec 15 '12 at 17:15
    
@DavideGiraudo: Thanks. –  Mithun Dec 15 '12 at 17:22
    
Expand $(\varphi+h)^3$, and multiplying by $\varphi+h$, we can remove terms in $h^j$, $j\geqslant 2$. What is the linear part in $h$? –  Davide Giraudo Dec 15 '12 at 17:28
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1 Answer

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. –  Mithun Dec 15 '12 at 18:17
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