
sahayak maanak sthaan

karyatmak vishleshan mein, disk seey maanak sthaanon key nirmaan key dou tarikon koo alexander grothendic dwara paramanu operators aur paramanu sthaanon koo paribhaashit karane key liye vyavasthit roop seey niyojit kiya gayaa thaa. 
yadi disk sootra _ 1 bandha huaa haih iss mamle mein, sahayak maanak sthaan maanak key saath sootra _ 2 haye tou ack vidhi kaa upayog kiya jaataa haye. 
sootra _ 3 
doosari vidhi kaa upayog kiya jaataa haye yadi disk sootra _ 1 avashoshit hoe rhaa haih iss mamle mein, sahayak maanak sthaan bhagfal sthaan sootra _ 5 haye. 
yadi disk bandhi hui aur avashoshit donon haye tou dou sahayak maanak sthaan vihit roop seey samroop hain (topological vector space key roop mein aur maanak space key roop mein).
ack bandhi hui disk-banach disk dwara prerit.
iss lekh key dauran, sootra _ 6 ack vaastavik yaa jatil sadish sthaan hoga (jaruri naheen kii ack tv, abhee taka) aur sootra _ 1 sootra _ 8 mein ack disk hoga.
disk dwara prerit ardh-kakshiya sthaan.
maan leejiye kii sootra _ 6 ack vaastavik yaa jatil sadish sthaan hoga. sootra _ 11 key kissy bhee upasamuchchay sootra _ 1 key liye sootra _ 1 key "minkowski karyatmak" dwara paribhaashit kiya gayaa haih
maan leejiye kii sootra _ 6 ack vaastavik yaa jatil sadish sthaan hoga. sootra _ 6 key kissy bhee upasamuchchay sootra key liye iss tarah kii minkovsky karyatmak sootra _ 28 sootra _ 29 parr ack ardh-sootra haye sootra _ 30 koo darshaata haye. 
sootra _ 31
jise sootra _ 32 dwara prerit ardh-angiya sthaan kahaa jaataa haye, jahan yadi sootra _ 28 ack maanak haye tou ise sootra _ 34 dwara prerit maanak sthaan kahaa jaataa haye."
dhaarana (topology): sootra _ 35 sootra _ 36 dwara prerit seminorm topology seey sampann haye jise sootra _ 37 yaa sootra _ 38 dwara darshaaya jaayega.
mahatvapurna roop seey, yeh topology "poori tarah seey" sootra _ 11 kii beejganitiya sanrachana aur sootra _ 41 parr samanya topology seey upaji haye (kyonki sootra _ 28 koo sett sootra _ 1 aur scalar gunan kaa upayog karke paribhaashit kiya gayaa haye). yeh banach disk key adhyayan koo uchit thaharaata haye aur iss kaaran kaa hissa haye kii way paramanu prachaalak aur paramanu sthaanon key siddhant mein mahatvapurna bhoomika kyon nibhaate hain.
samaveshan maanchitra sootra _ 44 koo "canonical mapp" kahaa jaataa haye.
maan leejiye kii sootra _ 1 ack disk haye. 
phir sootra _ 46 taaki sootra _ 1 sootra _ 29 mein avashoshit hoe rhaa haye sootra _ 34 kii raikhik avadhi 
sootra _ 1 key sabhi sakaratmak scalar gunakon kaa sett sootra _ 50 sootra _ 53 parr sthaaniya roop seey uttal sthalvaigyaanik sadish sthaan topology sootra _ 37 key liye mool mein pados kaa aadhar banaata haye. 
sootra _ 2 mein disk sootra _ 1 kaa minkovsky karyatmak iss baat kii guaranti detaa haye kii sootra _ 28 achchi tarah seey paribhaashit haye aur sootra _ 53 parr ack seminorm banaata haye. 
iss ardh-roop dwara prerit sthaaniya roop seey uttal topology topology sootra haye _ 37 jise pahale paribhaashit kiya gayaa thaa.
disk kii paribhasha.
ack topological vector space formula _ 6 mein ack bounded disk formula _ 1 jaise kii formula _ 61 ack banach space haye jise banach disk, infracomplete, yaa formula _ 8 mein bounded completaint kahaa jaataa haye.
yadi yeh dikhaaya jaataa haye kii sootra _ 63 ack banach sthaan haye tou sootra _ 1 tv mein ack banach disk hogi jisamen sootra _ 1 ack baadhya upasamuchchay key roop mein hoga.
aisaa isaliye haye kyonki minkovsky karyatmak sootra _ 28 vishuddh roop seey beejganitiya shabdon mein paribhaashit haye. 
natijatan, yeh sawal kii kyaa sootra _ 61 ack banach sthaan banaata haye yaa naheen, kewal disk sootra _ 1 aur minkovsky karyatmak sootra _ 36 parr nirbhar haye aur kissy vishesh tvs topology parr naheen joe sootra _ 6 leey jaa sakata haye. 
iss prakaar aavashyakta yeh haye kii ack tvs sootra _ 6 mein ack banach disk sootra _ 6 kaa ack baadhya upasamuchchay hoe, yeh ekmatra guna haye joe ack banach disk kii topology koo usake tvs sootra _ 8 vaale topology seey jodata haye.
disk prerit ardh-vinyaasit rikt sthaan key guna.
bandhi hui disk
nimnalikhit parinam bataata haye kii banach disk koo kyon bandha honah aavashyak haye.
badahavaspan
space formula _ 61 taba hotaa haye jaba formula _ 28 ack maanak hoe, joe taba hotaa haye jaba formula _ 1 mein koee gair-tuchchh vector subspace naa hoe. 
vishesh roop seey, yadi sootra _ 6 parr ack hasdorf tvs topology maujood haye jaise kii sootra _ 1 sootra _ 6 mein bandha huaa haye tou sootra _ 28 ack maanak haye. 
ack udaaharan jahan sootra _ 30 koo sootra _ 82 key roop mein praapt kiya jaataa haye aur sootra _ 1 koo sootra _ 84-aksh honey diyaa jaataa haye.
jaalon kaa abhisaran
maan leejiye kii sootra _ 1 sootra _ 6 mein ack disk haye jaise kii sootra _ 30 hasdorff haye aur sootra _ 88 koo sootra _ 89 mein ack shuddh honey dein. 
phir sootra 30 mein sootra 90 yadi aur kewal tabhi jaba vaastavik sankhyaon kaa ack shuddh sootra maujood hoe jaise kii sootra 93 aur sabhi sootra 95 key liye sootra 94; 
isake alawa, iss mamle mein yeh maanaa jaayega kii sabhi sootra key liye samanyata key nuksan key binaa sootra _ 96 _ 97
disk-prerit rikt sthaan key beach sambandh
yadi sootra 98 haye tou sootra 99 aur sootra 100 sootra 101 parr haye tou nimnalikhit nirantar raikhik maanchitra koo paribhaashit karein.
yadi sootra _ 102 aur sootra _ 1 sootra _ 105 key saath sootra _ 6 mein disk hain tou samavesh maanchitra sootra _ 106 koo sootra _ 107 key "vihit samavesh" sootra _ 89 mein kahein.
vishesh roop seey, upa-sthaan topology joe sootra _ 109 sootra _ 61 seey viraasat mein praapt hotaa haye, sootra _ 111 key seminorm topology seey kamjor haye.
disk bund ikai gend key roop mein
disk sootra _ 1 sootra _ 61 kaa ack bund upasamuchchay haye yadi aur kewal tabhi yadi sootra _ 1 seminorm sootra _ 28 kii bund ikai gend haye; arthaat, 
sootra _ 116 
yadi sootra _ 1 ack vector space sootra _ 6 mein ack disk haye aur yadi sootra _ 2 parr ack tvs topology sootra _ 119 maujood haye jaise kii sootra _ 1 sootra _ 122 kaa ack bund aur bandha upasamuchchay haye tou sootra _ 1 sootra _ 61 (yaani, sootra _ 125) kii bund ikai gend haye (pramaan key liye futnote dekhein).
ack banach disk key liye paryaapt sthitiyaan.
nimnalikhit pramey kaa upayog yeh sthaapit karane key liye kiya jaa sakata haye kii sootra _ 61 ack banach space haye. 
ack baar jaba yeh sthaapit hoe jaataa haye, tou sootra _ 1 kissy bhee tv mein ack banach disk hogi jisamen sootra _ 1 bandha hoga.
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iska tatparya haye kii kissy bhee sootra key liye _ 129
sootra _ 130
taaki vishesh roop seey, sootra _ 131 lene seey yeh usa sootra _ 132 kaa anusaran kartaa haye joe sootra _ 133 mein nihit haye. 
choonki sootra _ 37 sootra _ 135 seey behtar haye sootra _ 132 sootra _ 137 mein ack kauchi anukram haye. 
sabhi sootra key liye _ 138 sootra _ 139 sootra _ 137 kaa kramik roop seey poorna upasamuchchay haye. 
vishesh roop seey, yeh sootra 141 key liye sach haye isaliye kuchh sootra _ 142 maujood haye jaise kii sootra _ 143 sootra _ 137 mein haye.
choonki sabhi sootra key liye sootra _ 145 _ 129 sootra _ 147 koo taya karke aur seema (sootra _ 148 mein) koo sootra _ 149 key roop mein lete hue yeh pratyek sootra _ 151 key liye sootra _ 150 kaa anusaran kartaa haye. 
iska tatparya haye kii sootra _ 152 sootra _ 153 key roop mein joe sootra _ 155 mein thheek vahee sootra _ 143 kahata haye. 
yeh darshaata haye kii sootra _ 156 puuraa hoe gayaa haye.
≤ iss dhaarana kii anumati haye kyonki sootra 132 ack metrick sthaan mein ack kauchi anukram haye (isaliye sabhi anuvarti kii seemaaen samaan hain) aur ack metrick sthaan mein ack anukram taba abhisaran kartaa haye jaba pratyek anuvarti mein ack upa-anukram hotaa haye joe abhisaran kartaa haye.
dhyaan dein kii bhale hii sootra _ 1 kissy bhee hasdorf tv kaa ack baadhya aur kramik roop seey poorna upasamuchchay naheen haye, phir bhee koee yeh nishkarsh nikaalane mein saksham hoe sakata haye kii sootra _ 61 ack banach space haye, iss pramey koo kissy disk sootra _ 160 koo santusht karane key liye lagu karke. 
sootra _ 161
kyonki sootra _ 162
uparokt pramey key nimnalikhit parinam hainh
maan leejiye kii sootra _ 1 ack tv sootra _ 8 mein ack bounded disk haye.
banach disk key guna.
sootra _ 6 koo ack tv honey dein aur sootra _ 1 koo sootra _ 8 mein ack bounded disk honey dein.
yadi sootra _ 1 sthaaniya roop seey uttal sthaan sootra _ 6 mein ack bounded banach disk haye aur yadi sootra _ 175 sootra _ 6 mein ack bairal haye tou sootra _ 175 sootra _ 1 koo avashoshit kartaa haye (yaani, ack sankhya sootra _ 180 haye joe sootra _ 180 haye).
yadi sootra _ 181 sootra _ 6 mein mool kaa ack uttal santulit bund pados haye tou sabhi pados sootra _ 183 kaa sangrah jahaan sootra _ 179 sakaratmak vaastavik sankhyaon parr hotaa haye, sootra _ 8 parr ack topological vector space topology koo prerit kartaa haye jaba sootra _ 6 mein yeh topology hoti haye, tou ise sootra _ 187 dwara darshaaya jaataa haye kyonki yeh topology aavashyak roop seey hasdorff naheen haye aur naa hii poorna haye, iss sootra _ 189 key puuraa honey koo sootra _ 189 dwara darshaaya jaataa haye taaki sootra _ 189 ack poorna hasdorff space haye aur sootra _ 191 iss sthaan parr ack maanak haye joe ack banach space mein 189 banaata haye. sootra _ 193 sootra _ 194 kaa dhruv sootra _ 195 mein ack kamjor roop seey compact bounded equicontinues disk haye aur isaliye yeh avikasit haye.
yadi sootra _ 6 sthaaniya roop seey uttal tvs haye tou sootra _ 11 key pratyek baadhya upasamuchchay sootra _ 197 key liye sootra _ 6 mein ack baadhya disk sootra _ 1 maujood haye jaise kii sootra _ 201 aur sootra _ 6 aur sootra _ 30 donon sootra _ 204 parr ack hii upa-sthaan topology koo prerit karate hain.
ack radial disk-bhagfal dwara prerit.
maan leejiye kii sootra _ 6 ack topological vector space haye aur sootra _ 206 ack uttal santulit aur radial sett haye. 
phir sootra _ 207 sootra _ 8 parr kuchh sthaaniya uttal topology sootra _ 208 key liye mool mein ack pados aadhar haye. 
yeh tvs topology sootra _ 208 sootra _ 211 sootra _ 212 dwara gathit minkowski karyatmak dwara diyaa gayaa haye joe sootra _ 6 parr ack ardh-sootra haye jise sootra _ 214 dwara paribhaashit kiya gayaa haye. 
topology sootra _ 208 hasdorf yadi aur kewal tabhi jaba sootra _ 216 ack maanak haye, yaa samkaksh roop seey, yadi aur kewal tabhi jaba sootra _ 217 yaa samkaksh, jisake liye yeh paryaapt haye kii sootra _ 206 sootra _ 8 mein bandha hoe. 
topology sootra _ 208 koo hasdorf honey kii aavashyakta naheen haye lekin sootra _ 221 hasdorf haye. 
sootra _ 221 parr ack maanak sootra _ 223 dwara diyaa jaataa haye jahan yeh moolya vaastav mein chune gaye samatulya varg sootra _ 224 key pratinidhi seey swatantra haye. 
maanak sthaan sootra _ 225 koo sootra _ 226 dwara darshaaya jaataa haye aur isake puuraa honey koo sootra _ 227 dwara darshaaya jaataa haye.
yadi sootra _ 206 sootra _ 6 mein bandha huaa haye tou ardh-sootra sootra _ 230 ack maanak haye isaliye vishesh roop seey, sootra _ 231 
iss mamle mein, humm sootra _ 234 key bajaay sootra _ 226 koo vector space sootra _ 6 key roop mein lete hain taaki sanketan sootra _ 226 spasht hoe (chaahe sootra _ 226 ack radial disk dwara prerit sthaan koo darshaata haye yaa ack bounded disk dwara prerit sthaan).
sootra _ 221 parr bhagfal topology sootra _ 237 (sootra _ 6 key mool topology seey viraasat mein milaa) maanak topology kii tulanaa mein behtar (samanya roop seey, sakhti seey maheen) haye.
vihit maanchitra.
canonical mapp" bhagfal maanchitra sootra 240 haye joe nirantar hotaa haye jaba sootra _ 226 mein yaa tou maanak topology yaa bhagfal topology hoti haye.
yadi sootra _ 181 aur sootra _ 206 radial disk hain jaise kii sootra _ 244 tou sootra _ 245 isaliye ack nirantar raikhik adhikshep "canonical mapp" sootra _ 246 haye jise bhejane dwara paribhaashit kiya gayaa haye. 
samatulya varg sootra key liye sootra _ 247 _ 248 jahan koee yeh satyaapit curr sakata haye kii paribhasha samatulya varg sootra _ 249 key pratinidhi parr nirbhar naheen karati haye joe chunaa gayaa haye. 
iss canonical maanchitra mein maanak sootra _ 250 haye aur iska sootra _ 189 key liye ack advitiya nirantar canonical vistaar haye jise sootra _ 252 dwara darshaaya jaataa haye.
maan leejiye kii sootra 253 aur sootra 102 key atirikt sootra 256 key saath sootra 6 mein bandhi hui disk hain taaki sootra 257 aur samavesh sootra 258 ack nirantar raikhik maanchitra hoe. 
sootra _ 259 sootra _ 260 aur sootra _ 258 koo vihit maanchitra honey dein. 
phir sootra _ 262 aur sootra _ 263
ack bandhi hui radial disk dwara prerit.
maan leejiye kii sootra _ 264 ack bounded radial disk haye. 
choonki sootra _ 264 ack bounded disk haye, yadi sootra _ 266 haye tou humm maanak sootra _ 268 key saath sahayak maanak sthaan sootra _ 35 banaa sakate hain; kyonki sootra _ 264 radial haye, sootra _ 270 haye. 
choonki sootra _ 264 ack radial disk haye, yadi sootra _ 272 haye tou humm ardh-sootra sootra _ 274 key saath sahayak ardh-kramit sthaan sootra _ 221 banaa sakate hain; kyonki sootra _ 264 bandha huaa haye, yeh ardh-sootra ack maanak aur sootra _ 276 haye isaliye sootra _ 277 haye. 
iss prakaar, iss mamle mein inn dou alag-alag tarikon seey utpaadit dou sahayak maanak sthaanon kaa parinam ack hii maanak sthaan mein hotaa haye.
dwait.
maan leejiye kii sootra _ 278 sootra _ 195 mein ack kamjor bund samvarti disk haye (iska tatparya haye kii sootra _ 278 kamjor roop seey compact haye) aur
sootra _ 281
sootra kaa dhruviya honah _ 282 
kyonki dwidhruvi pramey dwara sootra _ 283, yeh bataata haye kii ack nirantar raikhik karyatmak sootra _ 284 sootra _ 285 seey sambandhit haye yadi aur kewal tabhi yadi sootra _ 284 sootra _ 287 key nirantar dohare sthaan seey sambandhit haye jahaan sootra _ 288 sootra _ 290 dwara paribhaashit sootra _ 181 kaa minkovsky karyatmak haye.
sambandhit avadhaaranaaen.
tv mein ack disk koo "infraborniverus" kahaa jaataa haye yadi yeh sabhi banach disk koo avashoshit curr letee haye.
dou tvss key beach ack raikhik maanchitra koo "infrabounded" kahaa jaataa haye yadi yeh banach disk koo bounded disk mein mapp kartaa haye.
teji seey abhisaran.
ack anukram sootra _ 291 koo ack tv sootra _ 6 mein ack bindu sootra _ 293 key liye "tej abhisari" kahaa jaataa haye yadi ack banach disk sootra _ 1 maujood haye jaise kii sootra _ 84 aur anukram donon sootra _ 2 mein nihit hain aur sootra _ 298 mein sootra _ 297 haye.
pratyek tej abhisari anukram mackie abhisari hotaa haye.