▁theorem ▁of ▁functional ▁analysis . ▁Very ▁roughly , ▁it ▁states ▁that ▁most ▁well - be hav ed ▁norm ed ▁spaces ▁are ▁sub spaces ▁of ▁the ▁space ▁of ▁continuous ▁paths . ▁It ▁is ▁named ▁after ▁Stefan ▁Ban ach ▁and ▁Stanisław ▁Maz ur . ▁ ▁Stat ement ▁of ▁the ▁theorem ▁Every ▁real , ▁separ able ▁Ban ach ▁space ▁ ▁is ▁is omet r ically ▁is omorphic ▁to ▁a ▁closed ▁sub space ▁of ▁, ▁the ▁space ▁of ▁all ▁continuous ▁functions ▁from ▁the ▁unit ▁interval ▁into ▁the ▁real ▁line . ▁ ▁Comment s ▁On ▁the ▁one ▁hand , ▁the ▁Ban ach – M az ur ▁theorem ▁seems ▁to ▁tell ▁us ▁that ▁the ▁seem ingly ▁vast ▁collection ▁of ▁all ▁separ able ▁Ban ach ▁spaces ▁is ▁not ▁that ▁vast ▁or ▁difficult ▁to ▁work ▁with , ▁since ▁a ▁separ able ▁Ban ach ▁space ▁is ▁" only " ▁a ▁collection ▁of ▁continuous ▁paths . ▁On ▁the ▁other ▁hand , ▁the ▁theorem ▁tells ▁us ▁that ▁ ▁is ▁a ▁" re ally ▁big " ▁space , ▁big ▁enough ▁to ▁contain ▁every ▁possible ▁separ able ▁Ban ach ▁space . ▁ ▁Non - separ able ▁Ban ach ▁spaces ▁cannot ▁embed ▁is omet r ically ▁in ▁the ▁separ able ▁space ▁, ▁but ▁for ▁every ▁Ban ach ▁space ▁, ▁one ▁can ▁find ▁a ▁compact ▁Haus dor ff ▁space ▁ ▁and ▁an ▁is ometric ▁linear ▁embedding ▁ ▁of ▁ ▁into ▁the ▁space ▁ ▁of ▁scalar ▁continuous ▁functions ▁on ▁. ▁The ▁simplest ▁choice ▁is ▁to ▁let ▁ ▁be ▁the ▁unit ▁ball ▁of ▁the ▁continuous ▁dual ▁, ▁equ ipped ▁with