
karya nishchit qaraa

ganitamadhye, visheshtah mapan siddhantat, sanch karya heey eka karya aahe jyache kshetra heey dilelya kahee sanchanchya upasanchanche kutumb aahe aanee teey (sahasa) tyachi moolye vistarit vastavik sankhya reshechya sutramadhye ghete jyamadhye vastavik sankhyanche sootra _ 2 aanee sootra _ 3 asatat.
sanch karya samanyatah kahee prakare upasanchay karnyache uddisht thevate. mojmap hii sanch karye mojanyachi vaishishtyapurn udaharane aahet. tyamule, 'maap' chaa ganiteey arth aanee tyacha samanya bhashetil arth yanchyatil gondhal talhanyasathi 'sanch karya' haah shabd anekada vaparla jaato.
vyakhya.
jarr sootra 4 heey sootra 5 vareel sanchanche kutumb asel (mhanje sootra 6 jethe sootra 7 powerset darshavate) tarr a heey kshetra sootra 4 aanee kodomen sootra 10 saha eka karya sootra 8 aahe kinwa, kahivela, kodomen heey tyaeivaji kahee vector space asate, jase qee vector upay, jatil upay aanee prakshepan-moolya mojmap. 
sanch karyachya karyakshetrat konatehi sankhya gunadharm asoo shakatat; samanyatah aadhalanare gunadharm aanee kutumbanche varg khaleel taktyat soochibaddha aahet.
sarvasadharanpane, asey gruheet dharale jaate qee sootra _ 11 heey serve sootra _ 12 sathee nehmi changlya prakare paribhashit kele jaate kinwa samatulya aahe, teey sootra _ 8 heey sootra _ 14 aanee sootra _ 15 yaa donhi moolyanna moolya mhanoon ghet nahee. haah lekh yapudhe heey gruheet dharel; jaree vaikalpikaritya, khalee dilelya serve vyakhya tyaeivaji "jevha jevha berries/malika paribhashit kelee jaate" asha vidhanandware patra tharavata yeteel. heey kadhikadhi vaja karoon kele jaate, jase qee khaleel parinamansah, jeey jevha jevha sootra _ 8 maryaditpane yojak asate tevha dharan karateh 
null sanch
21 yaa sanch sutrala a (sootra _ 8 chya sandarbhat) kinwa fakt sootra _ 23 asey mhanatat. 
jevha jevha sootra _ 8 heey sootra _ 25 kinwa sootra _ 26 chya samaan nasate tevha samanyatah asey dekhil gruheet dharale jaate keeh
bhinnata aanee vastuman
sootra _ 27 aahe
sootra _ 28
jethe sootra _ 29 heey nirpeksh moolya darshavate (kinwa adhik samanyatah, sootra _ 8 heey vector-moolya (ardh) manak jaaget asalyas teey manak kinwa ardharoop darshavate). 
31 heey sootra gruheet dharalyas sootra 32 laa sootra 8 mhanatat aanee sootra 34 laa sootra 35 mhanatat. 
jarr pratyek sootra _ 36 sathee moolya sootra _ 37 asel (jyacha vyakhyenusar arth aahe sootra _ 38 aanee sootra _ 39; eka asey aahe jeey sootra _ 40 kinwa sootra _ 14 chya barobariche aahe) tarr seet function mhanatat. 
pratyek maryadit sanch karyamadhye maryadit vastuman asane aavashyak aahe. 
sanch karyanche samanya gunadharm.
sootra _ 4 varr eka sanch karya sootra _ 8 asey mhatale jaate
aniyantrit rakma
samanyikrut malikevaril yaa lekhachya vibhagat varnan kelyapramane, aniyantrit anukramanika sanch sutradvare anukramit kelelya vastavik sankhyanchya konatyahi kautumbik sutrasathi _ 44, tyanchya berries sutrala _ 46 hii maryadit aanshik berries sootra _ 47 chya nivval maryada mhanoon paribhashit karane shakya aahe, jethe domain sootra _ 48 heey sootra _ 49 dware nirdeshit kele jaate. 
jevha jevha heey jaale ekatra yete tevha tyachi maryada chinhanchya sutradvare darshavili jaate, tarr jarr heey jaale tyaeivaji sootra _ 51 madhye valale tarr heey sootra _ 52 lihoon darshavile jaau shakate. 
rikamya sanchavaril konatihi berries shoonya mhanoon paribhashit kelee jaate; mhanje, jarr sootra _ 53 asel tarr vyakhyenusar sootra _ 54.
udaharanarth, jarr pratyek sutrasathi sootra _ 55 asel tarr sootra _ 57 
aanee heey sootra _ 58 dakhavale jaau shakate. 
jarr sootra _ 59 asel tarr samanyikrut malika sootra _ 46 sootra _ 2 madhye ekatra yete jarr aanee fakt jarr sootra _ 62 nehamichya arthane binshartpane (kinwa samatulya, purnapane ekatra hote) ekatra yete. 
jarr samanyikrut malika sootra _ 46 sootra _ 2 madhye ekatra aale tarr sootra _ 65 aanee sootra _ 66 donhi sootra _ 2 chya ghatakanmadhye dekhil ekatra yetat aanee sanch sootra _ 68 heey aavashyakpane mojanyayogya aahe (mhanje, ektar maryadit kinwa mojanyayogya anant); sootra _ 2 konatyahi manak jaagene badalalyas heey kharey rahate. 
heey asey aahe qee samanyikrut malika sootra _ 46 sathee sootra _ 2 kinwa sootra _ 72 madhye ektrit honyasathi heey aavashyak aahe qee serve parantu jastit jaast mojmapyogyapane anek sootra _ 73 heey sootra _ 74 chya barobariche asateel, jyacha arth assa aahe qee sootra _ 75 hii jastit jaast mojmapyogyapane anek shunyetar sandnyanchi berries aahe. 
vegalya prakare, jarr sootra _ 68 heey aganit asel tarr samanyikrut malika sootra _ 46 ekatra yet nahee. 
thodakyat, vastavik sankhyanche swaroop aanee tyanchya topologymule, ektrit honanya vastavik sankhyanchya pratyek samanyikrut malika (aniyantrit sanchadvare anukramit) mojata yenyajogya anek vastavik sankhyanchya samanya purnapane abhisari maliket qamee kelee jaau shakate. tyamule mapan siddhantachya sandarbhat, asankhya sanch aanee samanyikrut malikancha vichar kelyane farsa fayda hott nahee. visheshtah, mhanunach 'gananiy yogatmak' chee vyakhya phaar kvachitach sootra 4 madheel (aanee nehamichya gananiy malika sootra 80) gananiyapane anek sanch sootra 78 pasoon aniyantritpane anek sanch sootra 81 (aanee samanyikrut malika sootra 82) paryant vistarit kelee jaate.
aateel mojmap, bahya mojmap aanee itar gunadharm.
eka sanch karya sootra _ 8 heey/samadhankarak asalyache mhatale jaate
jarr binary operation sootra _ 84 paribhashit kele asel, tarr eka sanch karya sootra _ 8 asey mhatale jaate
topologieshy sambandhit vyakhya.
jarr sootra 86 heey sootra 5 vareel topology asel tarr eka sanch karya sootra 8 asey mhatale jaateh
sanch karyanmadhil sambandh.
jarr sootra _ 8 aanee sootra _ 90 hii sootra _ 91 vareel done sanch karye asateel tarah
udaharane.
sanch karyanchya udaharananmadhye khaleel goshtincha samavesh hotoh
sootra _ 97 vareel jorden mojmap heey sootra _ 98 chya serve jorden mojata yenyajogya upasanchanchya sanchavar paribhashit kelele eka sanch karya aahe, jeey tyachya jorden mojamapasathi jorden mojata yenyajogya sanchala pathavate.
lebsgue mojmap.
sootra 99 vareel lebsgue mojmap heey eka sanch karya aahe jeey lebsgue sootra 100-bijganitashi sambandhit asalelya vastavik sankhyanchya pratyek sanchala nakaratmak nasaleli vastavik sankhya niyukt karate. 
tyachi vyakhya vastavik sankhyanchya serve antaranchya sanch sootra _ 101 pasoon suru hote, jeey sootra _ 102 varche ardh bijganit aahe. 
pratyek madhyantar sutrala niyukt karnare karya _ 103 tyache sootra _ 104 heey eka maryaditpane yogatmak sanch karya aahe (spashtapane, jarr sootra _ 103 madhye antim bindu sootra _ 106 asel tarr sootra _ 107). 
heey sanch karya sootra _ 108 vareel bahya mojamapasathi vistarit kele jaau shakate jeey anuvad-avyavharik sanch karya sootra _ 109 aahe jeey _ 110 heey upasanch sootra anantavar pathavate.
sootra _ 111 
jaree tyache nirbandh caratheodory nikash purn karananya serve upasanch sootra _ 112 chya-bijganitavar asale taree bahya mojmap heey gananiyaritya yogatmak nahee (aanee mhanoon teey mojmap nahee):
sootra _ 113
heey eka mojmap aahe jyala lebsgue mojmap mhanatat. 
jeevan sanch hii vastavik sankhyanchya mojata naan yenyajogya sanchanchi udaharane aahet.
anant-aayami jaaga.
anant-aayami lebsgue mojamapachya lekhat tapshilvar sangitalyapramane, anant-aayami vegale karata yenyajogya manak jagevarche ekmev sthanik maryadit aanee anuvad-bhinn borle mojmap heey kshullak mojmap aahe. tathapi, anant-aayami topological vector spacessver gausian upay paribhashit karane shakya aahe. gausian upayansathiche sanrachana pramey darshavite qee amoort winer antaral bandhkam haah moolatah vegale karata yenyajoga banach jaaget katekorpane sakaratmak gausian upay midvanyacha ekmev marg aahe.
maryaditpane yogatmak anuvad-avyavharik sanch karye.
sootra 114 vareel ekmev bhashantar-bhinn maap, domain sootra 115 saha, jeey sootra 99 chya pratyek sankshipt upasanchavar maryadit aahe, heey kshullak sanch karya sootra 117 aahe, jeey sootra 95 chya samaan aahe (mhanje, teey pratyek sootra 119 laa sootra 95 varr pathavate)
tathapi, jarr mojata yenyajogya jodani maryadit jodanimadhye kamkuvat jhaali tarr yaa gunadharmansah eka kshullak nasalele sanch karya astitvat aahe aanee yavyatirikt, kahee sootra _ 121 madhye dekhil moolya dile jaate, kharey tarr, sootra _ 99 itar konatyahi abelian gut sootra _ 123 neey badalale tarihi ashee kshullak nasaleli sanch karye astitvat asateel.
sanch karye vadhavane.
ardh-vibhajkanpasun aljebraparyant vistarlela.
samaja sootra 8 heey sootra 5 varchya ardh bijganit sootra 4 varche eka sanch karya aahe aanee chala 
sootra _ 127
jeey sootra 129 dware tayar kelelya sootra 5 vareel bijganit aahe 
bijganitache mool udaharan, jeey bijganit dekhil nahee, teey kutumb aahe.
sootra _ 130
sootra _ 131 varr jethe serve sootra _ 133 sathee sootra _ 132 mahattvache aahe, sootra _ 135 madheel done gair-kathor asamanata sootra _ 134 heey kathor asamanata sootra _ 136 neey badalale jaau shakat nahee kaaran ardh-vibhajkamadhye sampoorn antarnihit sanch sootra _ 137 asane aavashyak aahe, mhanje sootra _ 138 hii ardh-vibhajakanchi aavashyakta aahe (sootra _ 139 pramane).
jarr sootra _ 8 heey maryaditpane yojak asel tarr tyache sootra _ 143 pathavun paribhashit kelelya sootra _ 141 varr sanch karya sootra _ 141 sathee eka adviteey vistar aahe (jethe sootra _ 144 heey soochit karate qee heey sootra _ 145 heey jodichya dishene vegale aahet):
sootra _ 146 
heey vistar sootra _ 141 dekhil maryaditpane yojak aselah konatyahi jodichya aadharavar visangati sutrasathi _ 148 
sootra _ 149 
jarr yaa vyatirikt sootra _ 8 heey vastavik-moolya asalele aanee ekatmata vistarit asel (jeey, visheshtah, sootra _ 8 heey nakaratmak naslyas asey hoil) tarr sootra _ 141 heey ekatmata aanee maryadit upadharmik aselah konatyahi sutrasathi _ 153 asey sootra _ 154
sootra _ 155
ringpasoon σ-aljebraparyant vistarat aahe.
jarr sootra _ 156 heey sanchanchya ringvareel (jase qee sanchanche bijganit) purv-mapan sootra _ 4 varr sootra _ 5 asel tarr sootra _ 8 laa sootra _ 129 dware tayar kelelya σ-bijganit sootra _ 161 varr mojmap sootra _ 160 paryant vistar aahe jarr sootra _ 8 heey σ-anant asel tarr haah vistar adviteey aahe.
haah vistar paribhashit karnyasathi, pratham sootra _ 8 chaa bahya mojmap sootra _ 165 paryant sootra _ 166 dware vistar qaraa. 
sootra _ 167 
aanee nantar teey sootra _ 165-mojata yenyajogya sanchanchya (mhanje, caratheodory-mojata yenyajogya sanch) sanch sootra _ 168 paryant maryadit qaraa, jeey serve sutrancha sanch aahe _ 170 asha prakare qee 
sootra _ 171 heey sootra _ 100-bijganit aahe aanee sootra _ 165 heey tyavar sigma-editive aahe, caratheodory lemma dware.
bahya upayanvar nirbandh ghalane.
jarr sootra _ 174 heey sanch sootra _ 91 vareel bahya mojmap asel, jethe (vyakhyenusar) kshetra heey aavashyakpane sootra _ 91 chey power seet sootra _ 7 asel, tarr upasanch sootra _ 170 laa 'kinwa' mhatale jaate, jarr teey khaleel goshtinche samadhan karateh 
sootra _ 17
jethe sootra _ 180 heey sootra _ 181 chey purak aahe. 
serve sootra _ 165-mojata yenyajogya upasanchanche kutumb heey σ-bijganit aahe aanee yaa kutumbasathi bahya mapan sootra _ 165 chey nirbandh heey eka mojmap aahe.
nondi.
purave
pratyek abhisari jaalyapramane, aanshik berries sootra 188 chey heey abhisari jaale a aahe, jyacha yaa vishisht jaalisathi (vyakhyenusar) arth assa aahe qee sootra 190 madheel moolchya pratyek shejarchya sootra 189 sathee sootra 103 chey eka maryadit upasanch sootra _ 191 astitvat aahe, jase qee 
serve maryadit superset sutransathi sootra _ 193 sootra _ 194
heey soochit karate qee pratyek sutrasathi _ 196 heey sootra (sootra _ 197 aanee sootra _ 198 gheuun). 
sootra _ 184 heey mapanyogya asalyane, tyache mool thikanavar mojata yenyajoge shejari aadhar sootra _ 200 aahe, jyache chhedanbindu aavashyakpane sootra _ 201 aahe (kaaran sootra _ 184 heey eka hasdorff tv aahe).
pratyek sakaratmak purnank sutrasathi _ 203 eka maryadit upasanch sootra _ 204 nivada, jase qee pratyek sutrasathi _ 205 sootra _ 206 
jarr sootra _ 207 heey sootra _ 208 chey asel tarr sootra _ 73 heey sootra _ 210 chey asel. 
asha prakare ganana karnyayogya sanch sootra _ 163 sootra _ 214 </refshee sambandhit nasalelya pratyek nirdeshank sootra _ 56 sathee sootra _ 211>