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Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$ \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}? $$

Yes, it is possible to define higher order Fréchet derivatives directly as bilinear functionals in $\mathscr{L}_2(F,\Bbb R)$ (the space of bounded bilinear functionals from $F$ to $\Bbb R$). This is shown by Ambrosetti and Prodi ([1], pp. 23-29), for example, and while leaving to them the details, we can simply say that $f:F\to\Bbb R$ is twice Fréchet differentiable at $\alpha\in F$ iff

• $F$ is one time Fréchet differentiable and
• $Df[\alpha+\beta](\gamma)-Df[\alpha](\gamma)= \mathfrak{B}[\alpha](\beta,\gamma) + o(\beta)$ where $o(\beta)/\Vert\beta\Vert_F\to 0$ as $\Vert\beta\Vert_F\to 0$ and $\mathfrak{B}\in\mathscr{L}\big(F,\mathscr{L}(F,\Bbb R)\big)\simeq\mathscr{L}_2(F,\Bbb R)$ (again, the proof of this fact is found in [1] §1.3, p. 23, where it is precisely show that the isomorphism between these two spaces is actually an isometry).

This can bee seen also by using the definition of Fréchet derivative you give, but it seems to me that, when dealing with functional derivatives of order greater than one, it overshadows the intrinsic clarity and familiarity of the concept. Precisely, by using the standard definition of Fréchet differentiablity (as given for example in [1] §1.1, p. 9, definition 1.1) and \eqref{1} we get $$ \begin{split} Df[\alpha+\varepsilon\beta](\gamma)-Df[\alpha](\gamma) &= \left\langle \frac{\delta f}{\delta \alpha}(\alpha+\beta),\gamma\right\rangle -\left\langle \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ & = \left\langle\frac{\delta f}{\delta \alpha}(\alpha+\beta)- \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ &=\left\langle\mathfrak{L}[\alpha](\beta)+ o(\beta),\gamma\right\rangle \end{split} $$ where $\mathfrak{L}\in\mathscr{L}\big(F,\mathscr{L}(F,E)\big)$.

Bibliography

[1] Ambrosetti, Antonio; Prodi, Giovanni, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34. Cambridge: Cambridge University Press, pp. viii+171 (1993), ISBN: 0-521-37390-5, MR1225101, ZBL0781.47046.

Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$ \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}? $$

Yes, it is possible to define higher order Fréchet derivatives directly as bilinear functionals in $\mathscr{L}_2(F,\Bbb R)$ (the space of bounded bilinear functionals from $F$ to $\Bbb R$). This is shown by Ambrosetti and Prodi ([1], pp. 23-29), for example, and while leaving to them the details, we can simply say that $f:F\to\Bbb R$ is twice Fréchet differentiable at $\alpha\in F$ iff

• $F$ is one time Fréchet differentiable and
• $Df[\alpha+\beta](\gamma)-Df[\alpha](\gamma)= \mathfrak{B}[\alpha](\beta,\gamma) + o(\beta)$ where $o(\beta)/\Vert\beta\Vert_F\to 0$ as $\Vert\beta\Vert_F\to 0$ and $\mathfrak{B}\in\mathscr{L}\big(F,\mathscr{L}(F,\Bbb R)\big)\simeq\mathscr{L}_2(F,\Bbb R)$ (again, the proof of this fact is found in [1] §1.3, p. 23, where it is precisely show that the isomorphism between these two spaces is actually an isometry).

This can bee seen also by using the definition of Fréchet derivative you give, but it seems to me that, when dealing with functional derivatives of order greater than one, it overshadows the intrinsic clarity and familiarity of the concept. Precisely, by using the standard definition of Fréchet differentiablity (as given for example in [1] §1.1, p. 9, definition 1.1) and \eqref{1} we get $$ \begin{split} Df[\alpha+\varepsilon\beta](\gamma)-Df[\alpha](\gamma) &= \left\langle \frac{\delta f}{\delta \alpha}(\alpha+\beta),\gamma\right\rangle -\left\langle \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ & = \left\langle\frac{\delta f}{\delta \alpha}(\alpha+\beta)- \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ &=\left\langle\mathfrak{L}[\alpha](\beta)+ o(\beta),\gamma\right\rangle \end{split} $$ where $\mathfrak{L}\in\mathscr{L}(F,E)$ is a linear bounded operator.

Bibliography

[1] Ambrosetti, Antonio; Prodi, Giovanni, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34. Cambridge: Cambridge University Press, pp. viii+171 (1993), ISBN: 0-521-37390-5, MR1225101, ZBL0781.47046.

Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$ \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}? $$

Yes, it is possible to define higher order Fréchet derivatives directly as bilinear functionals in $\mathscr{L}_2(F,\Bbb R)$ (the space of bounded bilinear functionals from $F$ to $\Bbb R$). This is shown by Ambrosetti and Prodi ([1], pp. 23-29), for example, and while leaving to them the details, we can simply say that $f:F\to\Bbb R$ is twice Fréchet differentiable at $\alpha\in F$ iff

• $F$ is one time Fréchet differentiable and
• $Df[\alpha+\beta](\gamma)-Df[\alpha](\gamma)= \mathfrak{B}[\alpha](\beta,\gamma) + o(\beta)$ where $o(\beta)/\Vert\beta\Vert_F\to 0$ as $\Vert\beta\Vert_F\to 0$ and $\mathfrak{B}\in\mathscr{L}\big(F,\mathscr{L}(F,\Bbb R)\big)\simeq\mathscr{L}_2(F,\Bbb R)$.

This can bee seen also by using the definition of Fréchet differentiability you give, but it seems to me that, when dealing with functional derivatives of order greater than one, it overshadows the intrinsic clarity and familiarity of the concept: $$ \begin{split} Df[\alpha+\varepsilon\beta](\gamma)-Df[\alpha](\gamma) &= \left\langle \frac{\delta f}{\delta \alpha}(\alpha+\beta),\gamma\right\rangle -\left\langle \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ & = \left\langle\frac{\delta f}{\delta \alpha}(\alpha+\beta)- \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ &=\left\langle\mathfrak{L}[\alpha](\beta)+ o(\beta),\gamma\right\rangle \end{split} $$

Bibliography

[1] Ambrosetti, Antonio; Prodi, Giovanni, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34. Cambridge: Cambridge University Press, pp. viii+171 (1993), ISBN: 0-521-37390-5, MR1225101, ZBL0781.47046.

Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$ \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}? $$

Yes, it is possible to define higher order Fréchet derivatives directly as bilinear functionals in $\mathscr{L}_2(F,\Bbb R)$ (the space of bounded bilinear functionals from $F$ to $\Bbb R$). This is shown by Ambrosetti and Prodi ([1], pp. 23-29), for example, and while leaving to them the details, we can simply say that $f:F\to\Bbb R$ is twice Fréchet differentiable at $\alpha\in F$ iff

• $F$ is one time Fréchet differentiable and
• $Df[\alpha+\beta](\gamma)-Df[\alpha](\gamma)= \mathfrak{B}[\alpha](\beta,\gamma) + o(\beta)$ where $o(\beta)/\Vert\beta\Vert_F\to 0$ as $\Vert\beta\Vert_F\to 0$ and $\mathfrak{B}\in\mathscr{L}\big(F,\mathscr{L}(F,\Bbb R)\big)\simeq\mathscr{L}_2(F,\Bbb R)$ (again, the proof of this fact is found in [1] §1.3, p. 23, where it is precisely show that the isomorphism between these two spaces is actually an isometry).

This can bee seen also by using the definition of Fréchet derivative you give, but it seems to me that, when dealing with functional derivatives of order greater than one, it overshadows the intrinsic clarity and familiarity of the concept. Precisely, by using the standard definition of Fréchet differentiablity (as given for example in [1] §1.1, p. 9, definition 1.1) and \eqref{1} we get $$ \begin{split} Df[\alpha+\varepsilon\beta](\gamma)-Df[\alpha](\gamma) &= \left\langle \frac{\delta f}{\delta \alpha}(\alpha+\beta),\gamma\right\rangle -\left\langle \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ & = \left\langle\frac{\delta f}{\delta \alpha}(\alpha+\beta)- \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ &=\left\langle\mathfrak{L}[\alpha](\beta)+ o(\beta),\gamma\right\rangle \end{split} $$ where $\mathfrak{L}\in\mathscr{L}\big(F,\mathscr{L}(F,E)\big)$.

Bibliography

[1] Ambrosetti, Antonio; Prodi, Giovanni, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34. Cambridge: Cambridge University Press, pp. viii+171 (1993), ISBN: 0-521-37390-5, MR1225101, ZBL0781.47046.

Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$ \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}? $$

Yes, it is possible to define higher order Fréchet derivatives directly as bilinear functionals $\mathscr{L}_2(F,\Bbb R)$. This is shown by Ambrosetti and Prodi ([1], pp. 23-29), for example, and while leaving to them the details, we can simply say that $f:F\to\Bbb R$ is twice Fréchet differentiable at $\alpha\in F$ iff

• $F$ is one time Fréchet differentiable and
• $Df[\alpha+\beta](\gamma)-Df[\alpha](\gamma)= \mathfrak{B}[\alpha](\beta,\gamma) + o(\beta)$ where $o(\beta)/\Vert\beta\Vert_F\to 0$ as $\Vert\beta\Vert_F\to 0$ and $\mathfrak{B}\in\mathscr{L}\big(F,\mathscr{L}(F,\Bbb R)\big)\simeq\mathscr{L}_2(F,\Bbb R)$.

This can bee seen also by using the definition of Fréchet differentiability you give, but it seems to me that, when dealing with functional derivatives of order greater than one, it overshadows the intrinsic clarity and familiarity of the concept: $$ \begin{split} Df[\alpha+\varepsilon\beta](\gamma)-Df[\alpha](\gamma) &= \left\langle \frac{\delta f}{\delta \alpha}(\alpha+\beta),\gamma\right\rangle -\left\langle \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ & = \left\langle\frac{\delta f}{\delta \alpha}(\alpha+\beta)- \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ &=\left\langle\mathfrak{L}[\alpha](\beta)+ o(\beta),\gamma\right\rangle \end{split} $$

Bibliography

[1] Ambrosetti, Antonio; Prodi, Giovanni, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34. Cambridge: Cambridge University Press, pp. viii+171 (1993), ISBN: 0-521-37390-5, MR1225101, ZBL0781.47046.

Now, I'd like to know how to define higher order derivatives of functional derivatives. In other words, suppose the Fréchet derivative of $f$ at $\alpha$, $Df(\alpha)$ is Fréchet differentiable at $\beta\in F$, is it possible to define $$ \dfrac{\delta^{2}f}{\delta \beta\delta\alpha}? $$

Yes, it is possible to define higher order Fréchet derivatives directly as bilinear functionals in $\mathscr{L}_2(F,\Bbb R)$ (the space of bounded bilinear functionals from $F$ to $\Bbb R$). This is shown by Ambrosetti and Prodi ([1], pp. 23-29), for example, and while leaving to them the details, we can simply say that $f:F\to\Bbb R$ is twice Fréchet differentiable at $\alpha\in F$ iff

• $F$ is one time Fréchet differentiable and
• $Df[\alpha+\beta](\gamma)-Df[\alpha](\gamma)= \mathfrak{B}[\alpha](\beta,\gamma) + o(\beta)$ where $o(\beta)/\Vert\beta\Vert_F\to 0$ as $\Vert\beta\Vert_F\to 0$ and $\mathfrak{B}\in\mathscr{L}\big(F,\mathscr{L}(F,\Bbb R)\big)\simeq\mathscr{L}_2(F,\Bbb R)$.

This can bee seen also by using the definition of Fréchet differentiability you give, but it seems to me that, when dealing with functional derivatives of order greater than one, it overshadows the intrinsic clarity and familiarity of the concept: $$ \begin{split} Df[\alpha+\varepsilon\beta](\gamma)-Df[\alpha](\gamma) &= \left\langle \frac{\delta f}{\delta \alpha}(\alpha+\beta),\gamma\right\rangle -\left\langle \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ & = \left\langle\frac{\delta f}{\delta \alpha}(\alpha+\beta)- \frac{\delta f}{\delta \alpha} (\alpha),\gamma\right\rangle\\ &=\left\langle\mathfrak{L}[\alpha](\beta)+ o(\beta),\gamma\right\rangle \end{split} $$

Bibliography

[1] Ambrosetti, Antonio; Prodi, Giovanni, A primer of nonlinear analysis, Cambridge Studies in Advanced Mathematics, 34. Cambridge: Cambridge University Press, pp. viii+171 (1993), ISBN: 0-521-37390-5, MR1225101, ZBL0781.47046.