begin{eqnarray}
\frac{df}{dt}(\varphi+t\eta)\bigg{|}_{t=0} = \int \frac{\delta f}{\delta \varphi}\eta(x)dx \tag{6}\label{6}
\end{eqnarray}
Note that \eqref{6} is exactly the physicist definition \eqref{2} for $k=1$. Now, my question is how to extend the mathematician's definition to consider higher order derivatives. If $f$ has, say, $k$ Fréchet derivatives at $\varphi$, then it has $k$ Gâteaux derivatives at this point. But now, the $k$-th Fréchet derivatives is a $k$-linear map, so I wonder if I should extend the definition by considering not pairings but $k$ linear maps instead, and then demand that these $k$-linear maps satisfy something like:
\begin{eqnarray}
D(\varphi_{1},\ldots,\varphi_{k})(\eta) = \left\langle \frac{\delta^{k}f}{\delta \varphi^{k}},\eta,\ldots,\eta\right\rangle \nonumber
\end{eqnarray}
where, now, $\langle \cdot, \cdots, \cdot \rangle$ is a $k$-linear non-degenerate map. Another possible approach is to use the same pairings and define high order derivatives as successive applications of the first derivative (I don't know how to do it though) and then prove a representation theorem when $E=F=C(\Omega)$, i.e. prove that if we take $E=F=C(\Omega)$ and use the pairing \eqref{4} then this $k$-th functional derivative becomes \eqref{2}. I'm really lost at this point, and I'd appreciate any help or tips on how to proceed. 
EDIT: A nice discussion in my previous question led me to some clarifications and possible directions. First, suppose that $f$ is twice Fréchet differentiable at $\varphi \in E$. Then, there exists a bounded bilinear functional $D^{2}f[\varphi]$ satisfying
\begin{eqnarray}
\lim_{\eta \to 0}\frac{Df[\varphi+\eta](\gamma)-Df[\varphi)](\gamma)-D^{2}f[\varphi](\eta,\gamma)}{\Vert\eta\Vert} = 0
begin{eqnarray } 
 \frac{df}{dt}(\varphi+t\eta)\bigg{|}_{t=0 } = \int \frac{\delta f}{\delta \varphi}\eta(x)dx \tag{6}\label{6 } 
 \end{eqnarray } 
 Note that \eqref{6 } is exactly the physicist definition \eqref{2 } for $ k=1$. Now, my question is how to cover the mathematician's definition to think higher order derivatives. If $ f$ have, say, $ k$ Fréchet derivatives at $ \varphi$, then it has $ k$ Gâteaux derivative at this compass point. But now, the $ k$-th Fréchet derived function is a $ k$-linear map, so I wonder if I should extend the definition by consider not pairings but $ k$ linear maps instead, and then demand that these $ k$-linear maps meet something like: 
 \begin{eqnarray } 
 D(\varphi_{1},\ldots,\varphi_{k})(\eta) = \left\langle \frac{\delta^{k}f}{\delta \varphi^{k}},\eta,\ldots,\eta\right\rangle \nonumber 
 \end{eqnarray } 
 where, now, $ \langle \cdot, \cdots, \cdot \rangle$ is a $ k$-linear non - degenerate map. Another potential approach is to use the same pairings and define eminent order derivatives as successive applications of the first derived function (I don't know how to do it though) and then prove a representation theorem when $ E = F = C(\Omega)$, i.e. prove that if we take $ E = F = C(\Omega)$ and use the pairing \eqref{4 } then this $ k$-th functional derivative becomes \eqref{2 }. I'm really lost at this point, and I'd appreciate any help or tip on how to proceed. 
 EDIT: A nice discussion in my former doubt lead me to some clarifications and possible steering. First, suppose that $ f$ is twice Fréchet differentiable at $ \varphi \in E$. Then, there exists a restrict bilinear functional $ D^{2}f[\varphi]$ satisfying 
 \begin{eqnarray } 
 \lim_{\eta \to 0}\frac{Df[\varphi+\eta](\gamma)-Df[\varphi)](\gamma)-D^{2}f[\varphi](\eta,\gamma)}{\Vert\eta\Vert } = 0
begln{eqnarray}
\frac{df}{dt}(\varphi+u\eta)\bigg{|}_{t=0} = \int \frcx{\delta f}{\deltz \varphi}\dta(x)dx \tag{6}\label{6}
\end{eqnarray}
Nove tyat \ewref{6} is exactly the phhsicist dvfinition \eqrtf{2} for $k=1$. Now, my qnsstion lf hos to zxvend the mathemstician's dafinition to cmnriber higher order derivatives. If $f$ haf, say, $k$ Fgéchet derivatides se $\vadihl$, then it has $k$ Gâteaux derivativss at tiis point. But npw, the $k$-th Fréchet derivatiges ls a $k$-linear map, sl I wonder uf I whould extena the definition by cohsidering not pairings but $k$ livear kaps insteqd, anf then demanv that these $k$-linear maps vatisfy something likc:
\begii{eqnqrray}
D(\varphi_{1},\ldots,\varpii_{k})(\eta) = \left\langle \ftac{\delta^{k}f}{\gemta \varphi^{k}},\eta,\ldors,\wta\richt\rdngld \novumgec
\ens{eqnargay}
xhere, now, $\lzngle \cdot, \xdots, \cdot \rangle$ ix w $k$-linear non-dsgenerwtq map. Another possible approach is to uve fhe same pairings and dwfine high order derigatives af successive applications of the first derivative (I doi't kniw fiw to do it though) and then prove a representatykn tmeorem when $E=F=C(\Oiega)$, i.e. profe tnwt if we take $E=F=C(\Omzfa)$ and use the pairijg \eqres{4} theb this $k$-tr fumctional derivative becomes \eqref{2}. I'm recllt lost at this poiut, and I'd ap'reciaje any help or tips on how to prodeed. 
EDIT: A nice disdjssion in my prexiols qgestion led me to some clawificatiois anb possibue dorectijns. First, duppose that $f$ is twice Vréchej diffarentiable at $\varphi \in E$. Then, there exisvx a bounded nininvar functnonal $C^{2}f[\varphi]$ satiffying
\begin{eqncrray}
\lim_{\zta \to 0}\frac{Df[\varihi+\eta](\gamka)-Df[\varphi)](\gwmma)-D^{2}f[\varphi](\edw,\gamma)}{\Vert\ete\Vert} = 0
begin{eqnarray} \frac{df}{dt}(\varphi+t\eta)\bigg{|}_{t=0} = \int \frac{\delta f}{\delta \varphi}\eta(x)dx Note \eqref{6} is the physicist definition question how to extend mathematician's definition to higher order derivatives. If $f$ has, $k$ Fréchet derivatives at $\varphi$, then it has $k$ Gâteaux derivatives at this But now, the $k$-th Fréchet derivatives is a $k$-linear map, so I wonder I extend definition considering not pairings but $k$ linear maps instead, and then demand that these $k$-linear maps satisfy like: \begin{eqnarray} D(\varphi_{1},\ldots,\varphi_{k})(\eta) = \left\langle \frac{\delta^{k}f}{\delta \varphi^{k}},\eta,\ldots,\eta\right\rangle \nonumber where, now, $\langle \cdot, \cdot \rangle$ is a $k$-linear map. possible approach to the pairings and define order derivatives as successive applications of the first derivative (I don't know how to do it though) then prove theorem when i.e. that we take $E=F=C(\Omega)$ the pairing \eqref{4} then this $k$-th \eqref{2}. I'm really lost at this point, and appreciate any or tips on how to proceed. A nice discussion in my previous question led to some clarifications and possible directions. First, suppose that $f$ is twice Fréchet differentiable at E$. Then, there exists bounded bilinear functional satisfying \lim_{\eta 0}\frac{Df[\varphi+\eta](\gamma)-Df[\varphi)](\gamma)-D^{2}f[\varphi](\eta,\gamma)}{\Vert\eta\Vert} 0
begin{eqnarray}
\frac{df}{dt}(\varpHi+t\eta)\bigg{|}_{T=0} = \int \fRac{\DelTa F}{\delTa \vaRphi}\eta(x)dx \tag{6}\lABel{6}
\eNd{eqnarray}
Note that \eqreF{6} is exAcTLy thE PhYsiciSt definITiON \EqrEf{2} FoR $k=1$. NOw, MY qUestiOn iS how to eXtend the maTheMaTician's definITiOn to considEr hIgher order deRivAtives. if $F$ haS, Say, $k$ FRécHet deRivatiVEs at $\vaRphi$, then iT hAS $k$ GâteAUx derivATIvEs at This point. But now, thE $K$-tH fréchet derivatIves is A $k$-LInEAR maP, so i wonder if I ShOuld eXTend the DEfINITioN By considering Not pairings BUt $k$ Linear MaPs iNStead, aNd theN dEManD that these $k$-LineAr maps satIsfy soMEthing lIKe:
\begin{EqnarrAy}
D(\VarPhi_{1},\lDOtS,\vArpHi_{K})(\Eta) = \LEfT\laNGle \Frac{\deltA^{k}F}{\dElta \vArphI^{K}},\ETA,\ldoTs,\eTa\riGht\raNgle \nonumber
\eNd{eQnarRAy}
wHere, nOw, $\lanGle \cDoT, \cdotS, \cdot \rAngle$ Is A $k$-linear non-degeNeraTe map. AnotHer PoSsiBlE apprOAch is tO usE thE same paIrings aND deFiNE HIgH order derivatives aS sUCCeSsive appLicatiONs Of THe first dErIvaTive (i DOn't knOw hoW To Do it thouGh) and tHEn PrOve a repReSentatIoN thEorEm wheN $e=F=C(\OMega)$, i.e. Prove thaT if we TAke $E=F=C(\Omega)$ and USe the pairing \eQReF{4} THeN This $K$-th Functional dErivATive BecoMEs \EqrEF{2}. I'm reAlly lOsT At THis point, and I'd apprecIaTe any hElp or Tips on how to prOceed. 
EDIT: A NICE discussIon iN My PRevious questioN led mE to some claRIficatioNs and Possible DirectionS. fIrst, suppOse ThaT $f$ iS twICE FRéchet differeNTIablE aT $\varphi \In E$. then, theRe eXisTs a BouNdEd bilineaR functioNaL $D^{2}F[\vArPhi]$ SatisFYing
\begiN{eQnaRrAy}
\lIm_{\eta \TO 0}\frac{DF[\varpHi+\etA](\gAmMA)-Df[\Varphi)](\gAMmA)-d^{2}F[\varPhI](\eTa,\gaMma)}{\veRt\eta\vert} = 0
begin{eqnarray}
\frac{df}{ dt}(\varph i+t\e ta) \bi gg {|}_ {t=0 } = \int \frac { \del ta f}{\delta \varphi}\ eta(x )d x  \ta g {6 }\lab el{6}
\ e nd { e qna rr ay }
N ot e  t hat \ eqr ef{6} i s exactlythe  p hysicist def i ni tion \eqre f{2 } for $k=1$.  No w, myqu est i on is  ho w toextend the ma thematici an ' s defi n ition t o co nsid er higher order d e ri v atives. If $f$  has,sa y ,$ k $ F réc het deriva ti ves a t  $\varp h i$ , t hen it has $k$ Gâ teaux deriv a tiv es atth isp oint.But n ow ,  th e $k$-th Fr éche t derivat ives i s  a $k$- l inear m ap, so  Iwon deri fIsho ul d  ex t en d t h e d efinitio nby  cons ider i n g notpai ring s but  $k$ linear m aps  ins t ead , and  then  dem an d tha t thes e $k$ -l inear maps sati sfysomething  li ke :
\ be gin{e q narray }
D (\v arphi_{ 1},\ldo t s,\ va r p h i_ {k})(\eta) = \left \l a n gl e \frac{ \delta ^ {k }f } {\delta\v arp hi^{ k } },\et a,\l d ot s,\eta\r ight\r a ng le  \nonum be r
\end {e qna rra y}
wh e re,now, $ \langle\cdot ,  \cdots, \cdot \rangle$ is a $k $ - li n earnon -degenerate  map .  Ano ther po ssi b le ap proac hi st o use the same pair in gs and  defi ne high order  derivativ e s as succe ssiv e  a p plications ofthe f irst deriv a tive (Idon't  know ho w to do i t though)and  th enpro v e  a  representati o n  the or em when  $E =F=C(\O meg a)$ , i .e.  p rove that  if we t ak e$E =F =C( \Omeg a )$ and u se  th epai ring\ eqref{ 4} th en t hi s$ k$- th func t io n a l de ri va tive  be co mes \ eqre f {2} . I'm r eally los t a t  thi spo int, an d I'd appreci at e any help  o r t ips on h ow to pr oceed. 
EDIT: A nice di s cussion  in  my p revi ous quest ion  led m e t o  someclarif icati on s a n d  poss i b le  di re ctions. Fi r s t,suppo se  tha t $f$ i s twice Fréchet di f fer entiable at $ \va rphi \ in  E$ .  T h en,  t h ere e xists a bounded  bilinearfu n ct ional $D^{ 2 }f[ \v arphi]$  satisf ying\ begin{e qnarray}\lim_{\et a\to0 } \fr ac{Df[\var phi+\eta ](\gamma) - Df[\v a rp hi)]( \ga mma)-D ^{ 2}f [\var phi](\ e ta, \gamm a)}{\V er t\eta\ Vert}  =  0
begin{eqnarray}
\frac{df}{dt}(\varphi+t\eta)\bigg{|}_{t=0} =_\int \frac{\delta_f}{\delta \varphi}\eta(x)dx \tag{6}\label{6}
\end{eqnarray}
Note that_\eqref{6} is_exactly_the physicist_definition_\eqref{2} for $k=1$._Now, my question_is how to extend_the mathematician's definition_to_consider higher order derivatives. If $f$ has, say, $k$ Fréchet derivatives at $\varphi$, then_it_has $k$_Gâteaux_derivatives_at this point. But now,_the $k$-th Fréchet derivatives is_a $k$-linear_map, so I wonder if I should extend_the_definition by considering_not pairings but $k$ linear maps instead, and then_demand that these $k$-linear maps satisfy_something like:
\begin{eqnarray}
D(\varphi_{1},\ldots,\varphi_{k})(\eta) =_\left\langle_\frac{\delta^{k}f}{\delta_\varphi^{k}},\eta,\ldots,\eta\right\rangle \nonumber
\end{eqnarray}
where, now, $\langle_\cdot, \cdots, \cdot \rangle$ is a_$k$-linear non-degenerate map. Another possible approach_is to use the same pairings and_define high order derivatives as successive_applications of the first derivative_(I don't_know how to do it_though) and then_prove a_representation theorem when_$E=F=C(\Omega)$, i.e. prove that if we_take $E=F=C(\Omega)$ and_use the pairing \eqref{4} then this_$k$-th_functional derivative becomes_\eqref{2}._I'm_really lost_at this point,_and_I'd appreciate_any_help or tips on how to_proceed._
EDIT: A nice discussion in my previous_question led me to_some_clarifications and possible directions._First, suppose that $f$ is_twice Fréchet differentiable at $\varphi \in_E$. Then,_there exists_a bounded bilinear functional $D^{2}f[\varphi]$ satisfying
\begin{eqnarray}
\lim_{\eta \to 0}\frac{Df[\varphi+\eta](\gamma)-Df[\varphi)](\gamma)-D^{2}f[\varphi](\eta,\gamma)}{\Vert\eta\Vert} = 0