▁analysis ▁and ▁non linear ▁analysis , ▁it ▁is ▁possible ▁to ▁define ▁the ▁derivative ▁of ▁a ▁function ▁between ▁two ▁Fr éch et ▁spaces . ▁ ▁This ▁notion ▁of ▁different iation , ▁as ▁it ▁is ▁Gate aux ▁derivative ▁between ▁Fr éch et ▁spaces , ▁is ▁significantly ▁we aker ▁than ▁the ▁derivative ▁in ▁a ▁Ban ach ▁space , ▁even ▁between ▁general ▁topological ▁vector ▁spaces . ▁ ▁Nevertheless , ▁it ▁is ▁the ▁weak est ▁notion ▁of ▁different iation ▁for ▁which ▁many ▁of ▁the ▁familiar ▁the or ems ▁from ▁calculus ▁hold . ▁ ▁In ▁particular , ▁the ▁chain ▁rule ▁is ▁true . ▁ ▁With ▁some ▁additional ▁constraints ▁on ▁the ▁Fr éch et ▁spaces ▁and ▁functions ▁involved , ▁there ▁is ▁an ▁analog ▁of ▁the ▁inverse ▁function ▁theorem ▁called ▁the ▁Nash – M oser ▁inverse ▁function ▁theorem , ▁having ▁wide ▁applications ▁in ▁non linear ▁analysis ▁and ▁differential ▁geometry . ▁ ▁Mathemat ical ▁details ▁ ▁Form ally , ▁the ▁definition ▁of ▁different iation ▁is ▁identical ▁to ▁the ▁Gate aux ▁derivative . ▁ ▁Specifically , ▁let ▁X ▁and ▁Y ▁be ▁Fr éch et ▁spaces , ▁U ▁ ⊂ ▁X ▁be ▁an ▁open ▁set , ▁and ▁F ▁: ▁U ▁→ ▁Y ▁be ▁a ▁function . ▁ ▁The ▁direction al ▁derivative ▁of ▁F ▁in ▁the ▁direction ▁v ▁ ∈ ▁X ▁is ▁defined ▁by ▁ ▁if ▁the ▁limit ▁exists . ▁ ▁One ▁says ▁that ▁F ▁is ▁continu ously ▁differenti able , ▁or ▁C 1 ▁if ▁the ▁limit ▁exists ▁for ▁all ▁v   ∈   X ▁and ▁the ▁mapping ▁D F : U ▁x ▁X ▁→