mathscr{L\!i\!n}(F, G) \quad\forall \varphi,\eta\in F\label{A}\tag{A}
$$
where $\varepsilon$ is a real parameter and $\mathscr{L\!i\!n}(F, G)$ is the vector space of all linear maps from $G$ to $F$ (including unbounded, and thus possibly discontinuous, ones).
If the functional derivative is also continuous respect to the topology of $G$ i.e. it belongs to $\mathscr{L}(F, G)\subsetneq\mathscr{L\!i\!n}(F, G)$, then it is a Fréchet derivative, thus $f$ is said to be Fréchet differentiable.
Let's define a (possibly multilinear) higher order functional derivative analogous to \eqref{A}: the classical idea seems to have been first explicitly stated by Fantappiè in [1], §5, pp. 513-514 (see also [2], chapter VI, §(25÷27), pp. 70-78). If $\varepsilon_1,\ldots\varepsilon_k$ are $k\in\Bbb N\setminus\{0\}$ real parameters, for all $\varphi,\eta_i,\ldots,\eta_k \in F$, the $k$-th order functional derivative is recursively defined as follows:
$$
\begin{split}
D f[\varphi](\eta_1)&=\bigg{[}\frac{\partial}{\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1]\bigg{]}_{\varepsilon_1 = 0}\\
D^2 f[\varphi](\eta_1,\eta_2)&=\bigg{[}\frac{\partial^2}{\partial\varepsilon_2\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1+\varepsilon_2 \eta_2]\bigg{]}_{\varepsilon_1, \varepsilon_2= 0} \\
&\qquad=\bigg{[}\frac{\partial}{\partial\varepsilon_2}Df[\varphi+\varepsilon_2 \eta_2](\eta_1)\bigg{]}_{\varepsilon_2 = 0}\\
\vdots\;\qquad &\:\,\vdots\quad\quad\qquad\qquad\vdots\\
D^k f[\varphi](\eta_1,\ldots,\eta_k) &=\left[\frac{\partial^k}{\partial\varepsilon_k
mathscr{L\!i\!n}(F, G) \quad\forall \varphi,\eta\in F\label{A}\tag{A } 
 $ $ 
 where $ \varepsilon$ is a real parameter and $ \mathscr{L\!i\!n}(F, G)$ is the vector space of all linear map from $ G$ to $ F$ (include unbounded, and thus possibly discontinuous, ones). 
 If the running derivative is also continuous respect to the regional anatomy of $ G$ i.e. it belong to $ \mathscr{L}(F, G)\subsetneq\mathscr{L\!i\!n}(F, G)$, then it is a Fréchet derivative, therefore $ f$ is say to be Fréchet differentiable. 
 Let's specify a (possibly multilinear) higher order running derivative analogous to \eqref{A }: the classical idea seems to have been first explicitly stated by Fantappiè in [ 1 ], § 5, pp. 513 - 514 (see besides [ 2 ], chapter VI, § (25÷27), pp. 70 - 78). If $ \varepsilon_1,\ldots\varepsilon_k$ are $ k\in\Bbb N\setminus\{0\}$ real parameters, for all $ \varphi,\eta_i,\ldots,\eta_k \in F$, the $ k$-th order running derivative is recursively defined as follows: 
 $ $ 
 \begin{split } 
 D f[\varphi](\eta_1)&=\bigg{[}\frac{\partial}{\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1]\bigg{]}_{\varepsilon_1 = 0}\\ 
 D^2 f[\varphi](\eta_1,\eta_2)&=\bigg{[}\frac{\partial^2}{\partial\varepsilon_2\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1+\varepsilon_2 \eta_2]\bigg{]}_{\varepsilon_1, \varepsilon_2= 0 } \\ 
 & \qquad=\bigg{[}\frac{\partial}{\partial\varepsilon_2}Df[\varphi+\varepsilon_2 \eta_2](\eta_1)\bigg{]}_{\varepsilon_2 = 0}\\ 
 \vdots\;\qquad & \:\,\vdots\quad\quad\qquad\qquad\vdots\\ 
 D^k f[\varphi](\eta_1,\ldots,\eta_k) & = \left[\frac{\partial^k}{\partial\varepsilon_k
matjscr{L\!i\!n}(F, G) \quad\forall \vavphi,\eta\in F\label{C}\rag{A}
$$
whxre $\varspsilon$ ks a real parameter and $\mathdce{L\!i\!n}(F, G)$ is the vector space of all lpnear mapw frin $G$ to $F$ (iidluding unbouhfed, cnv thus possibly discontingous, ones).
If tha wuuctional derivative is also continuots resprch to the topoljgy ps $G$ j.v. lt belongs to $\mathscr{L}(F, G)\subsetnsq\mathsbr{L\!i\!n}(F, G)$, then it os a Fréchet derivative, thud $f$ ls said to be Fréchft differenjjabjw.
Let's define a (possiblj multilinezr) higher order functional derixativz analogous ti \eetef{A}: the clawsicaj idea seems to have been forst explicitln staved vy Fantappiè in [1], §5, pp. 513-514 (vee also [2], chapter DI, §(25÷27), pp. 70-78). Ix $\rarepsilon_1,\ldots\varepsulin_k$ ate $k\it\Bbb B\segmihux\{0\}$ deal pwrajeters, for all $\varphi,\wta_i,\ldots,\eta_k \in F$, uhe $j$-th order fundtionaj qerivative is recursively defined as fonloss:
$$
\begin{split}
D f[\varphi](\etq_1)&=\bigg{[}\frac{\partial}{\partiwl\varepsijon_1}f[\varphi+\varepsilon_1 \eta_1]\bigg{]}_{\varepsilon_1 = 0}\\
D^2 f[\varphi](\ata_1,\ete_2)&=\bkgg{[}\yvwz{\pagtial^2}{\partial\varepsilon_2\partial\varepsilon_1}f[\varphy+\bateisilon_1 \eta_1+\varepsijon_2 \eta_2]\bigg{]}_{\fagelfilon_1, \varepsiuon_2= 0} \\
&\qquas=\bigg{[}\frac{\partial}{\pagtial\vatepsilin_2}Df[\varphy+\varrpsilon_2 \eta_2](\eta_1)\bigg{]}_{\varepsilob_2 = 0}\\
\vdots\;\qquab &\:\,\veots\quad\quad\qquad\qeuad\vdots\\
D^k f[\vatphi](\ets_1,\ldots,\eta_k) &=\left[\frac{\partnal^k}{\padtial\varepsllon_k
mathscr{L\!i\!n}(F, G) \quad\forall \varphi,\eta\in F\label{A}\tag{A} $$ where a parameter and G)$ is the maps $G$ to $F$ unbounded, and thus discontinuous, ones). If the functional derivative also continuous respect to the topology of $G$ i.e. it belongs to $\mathscr{L}(F, G)$, then it is a Fréchet derivative, thus $f$ is said to be differentiable. define (possibly higher order functional derivative analogous to \eqref{A}: the classical idea seems to have been first explicitly by Fantappiè in [1], §5, pp. 513-514 (see [2], chapter VI, §(25÷27), 70-78). If $\varepsilon_1,\ldots\varepsilon_k$ are $k\in\Bbb real for all \in the order functional derivative recursively defined as follows: $$ \begin{split} D f[\varphi](\eta_1)&=\bigg{[}\frac{\partial}{\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1]\bigg{]}_{\varepsilon_1 = 0}\\ D^2 f[\varphi](\eta_1,\eta_2)&=\bigg{[}\frac{\partial^2}{\partial\varepsilon_2\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1+\varepsilon_2 \eta_2]\bigg{]}_{\varepsilon_1, \varepsilon_2= 0} &\qquad=\bigg{[}\frac{\partial}{\partial\varepsilon_2}Df[\varphi+\varepsilon_2 \eta_2](\eta_1)\bigg{]}_{\varepsilon_2 \vdots\;\qquad &\:\,\vdots\quad\quad\qquad\qquad\vdots\\ f[\varphi](\eta_1,\ldots,\eta_k)
mathscr{L\!i\!n}(F, G) \quad\forall \varPhi,\eta\in F\lAbel{A}\Tag{a}
$$
whErE $\varEpsiLon$ is a real paraMEter And $\mathscr{L\!i\!n}(F, G)$ is the veCtor sPaCE of aLL lInear Maps froM $g$ tO $f$ (IncLuDiNg uNbOUnDed, anD thUs possiBly discontInuOuS, ones).
If the fuNCtIonal derivAtiVe is also contInuOus resPeCt tO The toPolOgy of $g$ i.e. it bELongs tO $\mathscr{L}(f, G)\SUbsetnEQ\mathscR{l\!I\!n}(f, G)$, thEn it is a Fréchet derIVaTIve, thus $f$ is said To be FrÉcHEt DIFfeRenTiable.
Let's DeFine a (POssibly MUlTILIneAR) higher order fUnctional deRIvaTive anAlOgoUS to \eqrEf{A}: thE cLAssIcal idea seeMs to Have been fIrst exPLicitly STated by fantapPiè In [1], §5, pP. 513-514 (see ALsO [2], cHapTeR vI, §(25÷27), pP. 70-78). if $\VarEPsiLon_1,\ldots\VaRePsiloN_k$ arE $K\IN\bbb N\SetMinuS\{0\}$ real Parameters, for All $\VarpHI,\etA_i,\ldoTs,\eta_K \in F$, ThE $k$-th oRder fuNctioNaL derivative is reCursIvely defiNed As FolLoWs:
$$
\begIN{split}
d f[\vArpHi](\eta_1)&=\biGg{[}\frac{\pARtiAl}{\PARTiAl\varepsilon_1}f[\varphI+\vAREpSilon_1 \eta_1]\Bigg{]}_{\vaREpSiLOn_1 = 0}\\
D^2 f[\varpHi](\Eta_1,\Eta_2)&=\bIGG{[}\frac{\PartIAl^2}{\Partial\vArepsiLOn_2\PaRtial\vaRePsilon_1}F[\vArpHi+\vArepsILon_1 \eTa_1+\varePsilon_2 \etA_2]\bigg{]}_{\VArepsilon_1, \varepSIlon_2= 0} \\
&\qquad=\bigg{[}\FRaC{\PArTIal}{\pArtIal\varepsilOn_2}Df[\VArphI+\varEPsIloN_2 \Eta_2](\etA_1)\bigg{]}_{\VaREpSIlon_2 = 0}\\
\vdots\;\qquad &\:\,\vdots\QuAd\quad\Qquad\Qquad\vdots\\
D^k f[\Varphi](\eta_1,\lDOTS,\eta_k) &=\lefT[\fraC{\PaRTial^k}{\partial\vaRepsiLon_k
mathscr{L\!i\!n}(F, G) \qu ad\forall\varp hi, \et a\ in F \lab el{A}\tag{A}
$ $ 
whe re $\varepsilon$ is arealpa r amet e rand $ \mathsc r {L \ ! i\! n} (F , G )$ is  thevec tor spa ce of alllin ea r maps from$ G$  to $F$ (i ncl uding unboun ded , andth usp ossib lydisco ntinuo u s, one s).
If th ef unctio n al deri v a ti ve i s also continuous re s pect to the to pology  o f  $ G $  i. e.it belongs  t o $\m a thscr{L } (F , G )\s u bsetneq\maths cr{L\!i\!n} ( F,G)$, t he n i t  is aFréch et der ivative, th us $ f$ is sai d to b e  Fréche t  differ entiab le. 
Le t'sd ef in e a  ( p oss i bl y m u lti linear)hi gh er or derf u n c tion alderi vativ e analogous t o \ eqre f {A} : the  clas sica lideaseemsto ha ve  been first exp lici tly state d b yFan ta ppièi n [1],  §5 , p p. 513- 514 (se e  al so [ 2 ],  chapter VI, §(25÷ 27 ) ,  p p. 70-78 ). If$ \v ar e psilon_1 ,\ ldo ts\v a r epsil on_k $  a re $k\in \Bbb N \ se tm inus\{0 \} $ real  p ara met ers,f or a ll $\v arphi,\e ta_i, \ ldots,\eta_k \ i n F$, the $k$ - th o rd e r fu nct ional deriv ativ e  isrecu r si vel y  defi ned a sf ol l ows:
$$
\begin{spli t} 
D f[\ varph i](\eta_1)&=\ bigg{[}\fr a c { \partial }{\p a rt i al\varepsilon_ 1}f[\ varphi+\va r epsilon_ 1 \et a_1]\big g{]}_{\va r e psilon_1  =0}\ \
D ^2f [ \v arphi](\eta_1 , \ eta_ 2) &=\bigg {[} \frac{\ par tia l^2 }{\ pa rtial\var epsilon_ 2\ pa rt ia l\v areps i lon_1}f[ \v arp hi +\v areps i lon_1\eta_ 1+\v ar ep s ilo n_2 \et a _2 ] \ bigg {] }_ {\va rep si lon_1 , \v a rep silon_2 = 0} \\
& \qq u ad=\ bi gg {[}\fra c{\partial}{\ pa rtial\vare ps ilo n_2}Df [ \ varphi+\ varepsilon_2 \eta_2](\e t a_1)\bi gg{ ]}_{\ vare psilon_2= 0 }\\
\v dot s \;\qqu ad &\: \,\vd ot s\q u a d\qua d \ qq uad \q quad\vdots \ \ 
D^ k f[\ va rphi ](\eta_ 1,\ldots,\eta_k) & = \le ft[\frac{\par tia l^k} { \ pa rti a l\ v are ps i lon _ k 
mathscr{L\!i\!n}(F, G)_\quad\forall \varphi,\eta\in_F\label{A}\tag{A}
$$
where $\varepsilon$ is a_real parameter_and_$\mathscr{L\!i\!n}(F, G)$_is_the vector space_of all linear_maps from $G$ to_$F$ (including unbounded,_and_thus possibly discontinuous, ones).
If the functional derivative is also continuous respect to the topology_of_$G$ i.e._it_belongs_to $\mathscr{L}(F, G)\subsetneq\mathscr{L\!i\!n}(F, G)$, then_it is a Fréchet derivative,_thus $f$_is said to be Fréchet differentiable.
Let's define a_(possibly_multilinear) higher order_functional derivative analogous to \eqref{A}: the classical idea seems_to have been first explicitly stated_by Fantappiè in_[1],_§5,_pp. 513-514 (see also_[2], chapter VI, §(25÷27), pp. 70-78)._If $\varepsilon_1,\ldots\varepsilon_k$ are $k\in\Bbb N\setminus\{0\}$ real_parameters, for all $\varphi,\eta_i,\ldots,\eta_k \in F$, the_$k$-th order functional derivative is recursively_defined as follows:
$$
\begin{split}
D f[\varphi](\eta_1)&=\bigg{[}\frac{\partial}{\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1]\bigg{]}_{\varepsilon_1_= 0}\\
D^2_f[\varphi](\eta_1,\eta_2)&=\bigg{[}\frac{\partial^2}{\partial\varepsilon_2\partial\varepsilon_1}f[\varphi+\varepsilon_1 \eta_1+\varepsilon_2 \eta_2]\bigg{]}_{\varepsilon_1, \varepsilon_2= 0}_\\
&\qquad=\bigg{[}\frac{\partial}{\partial\varepsilon_2}Df[\varphi+\varepsilon_2 \eta_2](\eta_1)\bigg{]}_{\varepsilon_2 =_0}\\
\vdots\;\qquad &\:\,\vdots\quad\quad\qquad\qquad\vdots\\
D^k_f[\varphi](\eta_1,\ldots,\eta_k) &=\left[\frac{\partial^k}{\partial\varepsilon_k