▁the ▁w * - top ology . ▁This ▁unit ▁ball ▁ ▁is ▁then ▁compact ▁by ▁the ▁Ban ach – A la og lu ▁theorem . ▁The ▁embedding ▁ ▁is ▁introduced ▁by ▁saying ▁that ▁for ▁every ▁, ▁the ▁continuous ▁function ▁ ▁on ▁ ▁is ▁defined ▁by ▁ ▁The ▁mapping ▁ ▁is ▁linear , ▁and ▁it ▁is ▁is ometric ▁by ▁the ▁H ahn – B an ach ▁theorem . ▁ ▁Another ▁general ization ▁was ▁given ▁by ▁Kle iber ▁and ▁Per vin ▁( 1 9 6 9 ): ▁a ▁metric ▁space ▁of ▁density ▁equal ▁to ▁an ▁infinite ▁cardinal ▁ ▁is ▁is ometric ▁to ▁a ▁sub space ▁of ▁, ▁the ▁space ▁of ▁real ▁continuous ▁functions ▁on ▁the ▁product ▁of ▁ ▁copies ▁of ▁the ▁unit ▁interval . ▁ ▁Str ong er ▁versions ▁of ▁the ▁theorem ▁Let ▁us ▁write ▁ ▁for ▁. ▁In ▁ 1 9 9 5 , ▁Luis ▁Rodríguez - P ia zza ▁proved ▁that ▁the ▁is ometry ▁ ▁can ▁be ▁chosen ▁so ▁that ▁every ▁non - zero ▁function ▁in ▁the ▁image ▁ ▁is ▁now here ▁differenti able . ▁Put ▁another ▁way , ▁if ▁ ▁consists ▁of ▁functions ▁that ▁are ▁differenti able ▁at ▁at ▁least ▁one ▁point ▁of ▁, ▁then ▁ ▁can ▁be ▁chosen ▁so ▁that ▁ ▁This ▁conclusion ▁applies ▁to ▁the ▁space ▁ ▁itself , ▁hence ▁there ▁exists ▁a ▁linear ▁map ▁ ▁that ▁is ▁an ▁is ometry ▁onto ▁its ▁image , ▁such ▁that ▁image ▁under ▁ ▁of ▁ ▁( the ▁sub space ▁consisting ▁of ▁functions ▁that ▁are ▁everywhere ▁differenti able ▁with ▁continuous ▁derivative ) ▁intersect s