)\big)\simeq \mathscr{L}\big(\times_{i=1}^{2}F,\Bbb R\big)\triangleq \mathscr{L}_2(F,\Bbb R)
$$ 
Earlier work. The key information in the formalism expressed by \eqref{B} is that it emphasizes the intrinsic multilinearity of higher order functional derivatives. In general function spaces, the concept of product of functions may not be defined therefore it couldn't be possible to define polynomials: therefore the straight notation 
$$
\bigg{[}\frac{\mathrm{d}^k}{\mathrm{d}\varepsilon^k}f[\varphi+\varepsilon \eta]\bigg{]}_{\varepsilon = 0} \triangleq D^k f[\varphi](\eta^k)
$$
could be misleading and overshadowing instead of shed light on the problems at hand. Nevertheless it is possible define abstract multilinear mappings and work with them. And while I have not being able to find earlier references for the notation \eqref{B} respect to the works of Fantappiè, Volterra is surely aware  of the necessity of working in a multilinear setting: his development of higher order functional derivative (remember that he works essentially in a Banach space setting) leads exactly to formula \eqref{2} ([3], §2.7, p. 102, formula (5) or [3], §29, p. 25) and to an extension of Taylor's theorem but his approach, while being essentially the same is much more involved, and requires several restriction on the functional $f$ ([2] §2.5, p.99, or [3], §27, pp. 22-24).
Life outside Banach spaces. As I briefly said in the main body of the question, formulas \eqref{B} were developed by Fantappiè to work with locally analytic functionals i.e. functionals defined on germs of holomorphic functions $ F=\mathscr{O}(\Bbb C)$ and $G=\Bbb C$. However, a more general and modern approach is developed by prof. Michor and described in the monograph linked in his answer: his approach is substantially based on the use of smooth/analytic curves with values in bornological spaces, i.e. on the use of functions of one real variable $c:\Bbb R\to B$ with $c\in C^\infty(\Bbb R, B)$ or $c\in C^\
) \big)\simeq \mathscr{L}\big(\times_{i=1}^{2}F,\Bbb R\big)\triangleq \mathscr{L}_2(F,\Bbb R) 
 $ $ 
 Earlier work. The key information in the formalism carry by \eqref{B } is that it stress the intrinsic multilinearity of higher order functional derivative. In general function space, the concept of product of functions may not be specify therefore it couldn't be possible to specify polynomials: therefore the square notation 
 $ $ 
 \bigg{[}\frac{\mathrm{d}^k}{\mathrm{d}\varepsilon^k}f[\varphi+\varepsilon \eta]\bigg{]}_{\varepsilon = 0 } \triangleq D^k f[\varphi](\eta^k) 
 $ $ 
 could be misleading and overshadowing instead of shed light on the problem at hand. Nevertheless it is possible define abstract multilinear mapping and work with them. And while I have not being able to find earlier references for the note \eqref{B } respect to the work of Fantappiè, Volterra is surely aware   of the necessity of working in a multilinear setting: his growth of higher order functional derivative (remember that he works essentially in a Banach space setting) leads exactly to formula \eqref{2 } ([ 3 ], § 2.7, p. 102, formula (5) or [ 3 ], § 29, p. 25) and to an extension of Taylor's theorem but his approach, while being essentially the same is much more byzantine, and necessitate several limitation on the running $ f$ ([ 2 ] § 2.5, p.99, or [ 3 ], § 27, pp. 22 - 24). 
 Life outside Banach spaces. As I briefly said in the main soundbox of the question, formulas \eqref{B } were developed by Fantappiè to work with locally analytic functionals i.e. functionals defined on microbe of holomorphic functions $ F=\mathscr{O}(\Bbb C)$ and $ G=\Bbb C$. However, a more general and modern approach is developed by prof. Michor and described in the monograph connect in his answer: his approach is substantially based on the habit of smooth / analytic bend with values in bornological spaces, i.e. on the use of functions of one real variable star $ c:\Bbb R\to B$ with $ c\in C^\infty(\Bbb R, B)$ or $ c\in C^\
)\big)\dimeq \mathscr{L}\big(\times_{i=1}^{2}F,\Nbb R\big)\triangleq \mathsrr{L}_2(F,\Bbb R)
$$ 
Earlidr work. The key information mn tye foemalism expressed by \edref{B} is nhat it enphawuzes the iifrinsic multimlnearntb of higher ordgr functionan derivatives. Hn gzneral function spaces, the concept os produvt of functions iay mjt bs defined therefore it couldn't be lossiblt to define polynokials: therefore the straigjt nltation 
$$
\bigg{[}\frac{\mahhrm{d}^k}{\mathrn{d}\vawwpsilon^k}f[\varohi+\varepsilon \eta]\bigg{]}_{\vzrepsilon = 0} \triangleq D^k f[\varphk](\eta^k)
$$
eould be miwlwadltg and overwhadoring instead of shed light pn the problemx av habd. Nevertheless it is possible define afstract mglcilinear mappings and wirk whth dhem. Qnd whjlx I have jot being abls to find eqrlier references fpr nne notation \sqref{B} rqspect to the works of Fantappiè, Volterrd ia surely aware  of the becessity of working ln a multylinear setting: his development of higher order fgnctikval dcrivxrige (remember that he works essentially in a Bagzcn xpace setting) jeads exactky tp formula \eqrew{2} ([3], §2.7, p. 102, fodmula (5) or [3], §29, p. 25) and to an gxtensuon of Tarlor'x theorem but his approach, qhile being vssebtially the same id much more invplved, and requires several rzstricfion on the functionzu $f$ ([2] §2.5, p.99, or [3], §27, pp. 22-24).
Uifv oudside Banach spaces. As I bwiefly samd in the makn bpdy of the questlon, formulas \eqref{B} were devepo[ed by Fanhappiè to work with locally analbvic functionaks i.e. functiouals dcfined on germs of holomorphie functijns $ W=\mathscr{O}(\Bgb C)$ anv $G=\Bbb C$. Howqver, a more gajeral and movern apprjach is eevelopda by prof. Michpr and described in the monograph linled jn his answer: hnr approach is sunstxntyaplb basqg on the use of rmouyh/anauytic curvew witn values in bornologhcal spaces, i.e. on the isc of funcjions of jne real varisble $c:\Bbb R\to B$ wiuh $c\in C^\iifty(\Bbn R, B)$ or $c\in C^\
)\big)\simeq \mathscr{L}\big(\times_{i=1}^{2}F,\Bbb R\big)\triangleq \mathscr{L}_2(F,\Bbb R) $$ Earlier key in the expressed by \eqref{B} intrinsic of higher order derivatives. In general spaces, the concept of product of may not be defined therefore it couldn't be possible to define polynomials: therefore straight notation $$ \bigg{[}\frac{\mathrm{d}^k}{\mathrm{d}\varepsilon^k}f[\varphi+\varepsilon \eta]\bigg{]}_{\varepsilon = 0} \triangleq D^k f[\varphi](\eta^k) $$ could be and instead shed on the problems at hand. Nevertheless it is possible define abstract multilinear mappings and work with And while I have not being able to earlier references for the \eqref{B} respect to the works Fantappiè, is surely of necessity working in a setting: his development of higher order functional derivative (remember that he works essentially in a Banach space leads exactly \eqref{2} ([3], p. formula or [3], §29, and to an extension of Taylor's approach, while being essentially the same is much involved, and several restriction on the functional $f$ §2.5, p.99, or [3], §27, pp. 22-24). Life Banach spaces. As I briefly said in the main body of the question, formulas \eqref{B} by Fantappiè to work locally analytic functionals functionals on of functions $ C)$ and $G=\Bbb C$. However, a more general and modern approach developed by prof. Michor and described in the monograph linked answer: approach is substantially on the use of curves values in bornological spaces, the of real $c:\Bbb B$ with $c\in C^\infty(\Bbb B)$ or $c\in C^\
)\big)\simeq \mathscr{L}\big(\times_{i=1}^{2}f,\Bbb R\big)\trIanglEq \mAthScR{L}_2(F,\BBb R)
$$ 
EArlier work. The kEY infOrmation in the formalism ExpreSsED by \eQReF{B} is tHat it emPHaSIZes ThE iNtrInSIc MultiLinEarity oF higher ordEr fUnCtional derivATiVes. In generAl fUnction spaceS, thE concePt Of pROduct Of fUnctiOns may NOt be deFined therEfORe it coULdn't be pOSSiBle tO define polynomialS: ThERefore the straiGht notAtIOn 
$$
\BIGg{[}\fRac{\Mathrm{d}^k}{\maThRm{d}\vaREpsilon^K}F[\vARPHi+\vARepsilon \eta]\biGg{]}_{\varepsiloN = 0} \TriAngleq d^k F[\vaRPhi](\eta^K)
$$
coulD bE MisLeading and oVersHadowing iNstead OF shed liGHt on the ProbleMs aT haNd. NeVErThEleSs IT is POsSibLE deFine abstRaCt MultiLineAR MAPpinGs aNd woRk witH them. And while i haVe noT BeiNg ablE to fiNd eaRlIer reFerencEs for ThE notation \eqref{B} RespEct to the wOrkS oF FaNtAppiè, vOlterrA is SurEly awarE  of the nECesSiTY OF wOrking in a multilineAr SETtIng: his deVelopmENt Of HIgher ordEr FunCtioNAL deriVatiVE (rEmember tHat he wORkS eSsentiaLlY in a BaNaCh sPacE settINg) leAds exaCtly to foRmula \EQref{2} ([3], §2.7, p. 102, formula (5) or [3], §29, P. 25) And to an extensIOn OF taYLor's TheOrem but his aPproACh, whIle bEInG esSEntiaLly thE sAMe IS much more involved, anD rEquireS seveRal restrictioN on the funcTIONal $f$ ([2] §2.5, p.99, or [3], §27, pP. 22-24).
LifE OuTSide Banach spacEs. As I Briefly saiD In the maiN body Of the queStion, formULAs \eqref{B} WerE deVelOpeD BY FAntappiè to worK WIth lOcAlly anaLytIc functIonAls I.e. fUncTiOnals defiNed on gerMs Of HoLoMorPhic fUNctions $ F=\MaThsCr{o}(\BbB C)$ and $g=\bbb C$. HoWever, A morE gEnERal And modeRN aPPRoacH iS dEvelOpeD bY prof. michOR anD descriBed in the mOnoGRaph LiNkEd in his Answer: his apprOaCh is substaNtIalLy baseD ON the use oF smooth/analytic curves wiTH values In bOrnolOgicAl spaces, i.E. on The use Of fUNctionS of one Real vArIabLE $C:\Bbb R\TO b$ wIth $C\iN C^\infty(\Bbb r, b)$ Or $c\In C^\
)\big)\simeq \mathscr{L}\b ig(\times_ {i=1} ^{2 }F, \B bb R \big )\triangleq \m a thsc r{L}_2(F,\Bbb R)
$$ 
E arlie rw ork. Th e key  inform a ti o n  in  t he  fo rm a li sm ex pre ssed by  \eqref{B}  is  t hat it empha s iz es the int rin sic multilin ear ity of  h igh e r ord erfunct ionald erivat ives. Inge n eral f u nctions p ac es,the concept of pr o du c t of functions  may n ot be d efi ned  therefore  i t cou l dn't be po s s i ble to define pol ynomials: t h ere fore t he  st r aightnotat io n  
$ $
\bigg{[}\ frac {\mathrm{ d}^k}{ \ mathrm{ d }\varep silon^ k}f [\v arph i +\ va rep si l on\ et a]\ b igg {]}_{\va re ps ilon= 0} \ t r iang leq  D^k  f[\v arphi](\eta^k )
$ $
co u ldbe mi slead ingan d ove rshado wingin stead of shed l ight  on the p rob le msat  hand .  Never the les s it is  possib l e d ef i n e  a bstract multilinea rm a pp ings and  workw it ht hem. And  w hil e Ih a ve no t be i ng  able to  finde ar li er refe re nces f or  th e n otati o n \e qref{B } respec t tot he works of Fa n tappiè, Volte r ra i ss urel y a ware  of th e ne c essi ty o f  w ork i ng in  a mu lt i li n ear setting: his de ve lopmen t ofhigher orderfunctional d e rivative  (re m em b er that he wor ks es sentiallyi n a Bana ch sp ace sett ing) lead s exactlytofor mul a \ e q re f{2} ([3], §2 . 7 , p.  1 02, for mul a (5) o r [ 3],  §2 9,p.  25) andto an ex te ns io nofTaylo r 's theor em  bu this  appr o ach, w hilebein ges s ent ially t h es a me i smu ch m ore  i nvolv ed,a ndrequire s several  re s tric ti on  on the  functional $ f$  ([2] §2.5 ,p.9 9, or[ 3 ], §27,pp. 22-24).
Life outsid e  Banach  sp aces.  AsI briefly  sa id inthe main b ody of  thequ est i o n, fo r m ul as\e qref{B} we r e  de velop ed  byFantapp iè to work with lo c all y analytic fu nct iona l s  i .e. fu n cti on a lsd e fined on germsof holomor ph i cfunctions$  F= \m athscr{ O}(\Bbb  C)$a nd $G=\ Bbb C$. H owever, a  m oreg e ner al and mod ern appr oach is d e velop e dby pr of.  Micho rand  desc ribedi n t he mo nograp hlinked  in h is  answer:  his approach is substa ntiall y bas edon the us e o f  sm ooth/anal ytic  curves wi thval ues i n b o rnolo gica l  s pac e s, i. e. o n  the useo ffun c t io ns of one r e a l  va riabl e $ c :\BbbR\to  B$ with $c\in C^ \ infty(\Bbb R,B)$o r  $c \in C^\
)\big)\simeq \mathscr{L}\big(\times_{i=1}^{2}F,\Bbb_R\big)\triangleq \mathscr{L}_2(F,\Bbb_R)
$$ 
Earlier work. The_key information_in_the formalism_expressed_by \eqref{B} is_that it emphasizes_the intrinsic multilinearity of_higher order functional_derivatives._In general function spaces, the concept of product of functions may not be defined_therefore_it couldn't_be_possible_to define polynomials: therefore the_straight notation 
$$
\bigg{[}\frac{\mathrm{d}^k}{\mathrm{d}\varepsilon^k}f[\varphi+\varepsilon \eta]\bigg{]}_{\varepsilon =_0} \triangleq_D^k f[\varphi](\eta^k)
$$
could be misleading and overshadowing instead of_shed_light on the_problems at hand. Nevertheless it is possible define abstract_multilinear mappings and work with them._And while I_have_not_being able to find_earlier references for the notation \eqref{B}_respect to the works of Fantappiè,_Volterra is surely aware  of the_necessity of working in a multilinear_setting: his development of higher_order functional_derivative (remember that he works_essentially in a_Banach space_setting) leads exactly_to formula \eqref{2} ([3], §2.7, p._102, formula (5)_or [3], §29, p. 25) and_to_an extension of_Taylor's_theorem_but his_approach, while being_essentially_the same_is_much more involved, and requires several_restriction_on the functional $f$ ([2] §2.5, p.99,_or [3], §27, pp._22-24).
Life_outside Banach spaces. As_I briefly said in the_main body of the question, formulas_\eqref{B} were_developed by_Fantappiè to work with locally analytic functionals i.e. functionals defined on_germs of holomorphic functions $ F=\mathscr{O}(\Bbb_C)$ and $G=\Bbb C$._However, a_more_general and modern_approach_is developed_by prof. Michor and described in the_monograph linked_in his answer: his approach is_substantially based on the_use_of smooth/analytic curves with values in_bornological spaces, i.e. on the use_of functions of one real_variable_$c:\Bbb_R\to B$ with $c\in C^\infty(\Bbb_R, B)$ or $c\in C^\