
paramanu antariksh

ganit mein, paramanu sthaan sthalvaigyaanik sadish sthaan hain jinhen parimit ayami euclidean sthaanon key saamaanyikaran key roop mein dekhaa jaa sakata haye aur unake kaee vaanchhaniya gunon koo saajha kiya jaa sakata haye. paramanu sthaan halanki hilbert sthaanon seey kaafee alag hain, joe seemit ayami euclidean sthaanon kaa ack aur saamaanyikaran haye. unhein alekjendar grothendic dwara pesh kiya gayaa thaa.
paramanu sthaanon parr topology koo seminorm key ack parivaar dwara paribhaashit kiya jaa sakata haye jisaki ikai genden aakar mein teji seey cumm hoe jaati hain. vector space jinake tatva kuchh arthon mein "chikane" hotey hain, way paramanu space hotey hain; ack paramanu space kaa ack vishisht udaaharan ack compact manifold parr chikane karyon kaa sett haye. sabhi parimit-ayami sadish sthaan paramanu hain. seemit ayami sthaanon koo chhodkar koee bhee aisaa sthaan naheen haye joe paramanu hoe. vyavhaar mein ack prakaar kaa iss parr abhisaran aksar sach hotaa haih yadi ack "praakritik roop seey honey vaalaa" sthalvaigyaanik sadish sthaan ack banach sthaan haye, tou ack achchi sambhavana haye kii yeh paramanu haye.
mool prernah schwartz kernell pramey.
paramanu sthaanon key adhikansh siddhant koo alekjendar grothendic dwara schwartz kernell pramey kii jaanch karate hue vikasit kiya gayaa thaa aur ise 2006 mein prakaashit kiya gayaa thaa. abb humm iss prerana kaa varnan karate hain. 
kissy bhee khule upasamuchchay sootra _ 1 aur sootra _ 2 key liye canonical maanchitra sootra _ 3, tii. vee. aiss. aiss. kaa ack samroopata haye (jahaan sootra _ 4 mein bandhe upasamuchchay parr samaan abhisaran kii topology haye) aur isake alawa, yeye donon sthaan canonical roop seey tii. vee. aiss-samroopata seey sootra _ 5 key liye hain (jahaan sootra _ 6 paramanu haye, yeh tensor utpaad ack saath injective tensor utpaad aur projective tensor utpaad haye). 
sankshep mein, schwartz kernell pramey bataata haye kih
sootra _ 7
jahan yeye sabhi tvs-isomorphism canonical hain. 
yeh parinam galat haye yadi koee antariksh sootra _ 8 koo sootra _ 9 seey badal detaa haye (joe ack prativartak sthaan haye joe apane swayam key majboot dohare sthaan key liye samroop haye) aur sootra _ 10 koo iss sootra _ 9 sthaan key dohare seey badal detaa haye. 
vitaran aur pareekshan karyon key sthaan key liye itna achchha parinam kyon haye, lekin hilbert space formula _ 9 (jise aam taur parr "sabse achchha" tvss mein seey ack maanaa jaataa haye) key liye kyon naheen? 
iss prashn ney grothendic koo paramanu sthaanon, paramanu maanchitron aur injective tensor utpaad kii khoj karane key liye prerit kiya.
jyaamiti seey preranaaen.
prerak udaharanon kaa ack aur samooh seedhe jyaamiti aur chikani manifold siddhant seey aataa haye-parishisht 2. chikani manifold sootra _ 13 aur sthaaniya roop seey uttal hasdorf topological vector sthaan diye gaye hain, tou paramanu sthaanon key nimnalikhit samaruupataaen hain.
sootra _ 16 key liye maanak tensor utpaadon kaa upayog ack vector sthaan key roop mein karate hue, falan sootra _ 17kanna koo sootra _ 19 key liye ack falan sootra _ 18 key roop mein vyakt naheen kiya jaataa haye, yeh ack udaaharan detaa haye joe yeh darshaata haye kii sett sootra _ 20 kaa sakht samavesh haye.
paribhasha.
iss khand mein paramanu antariksh kii kuchh adhik samanya paribhaashaaen soochibaddh hain. neeche dee gaee sabhi paribhaashaaen samatulya hain. dhyaan dein kii kuchh lekhak paramanu sthaan kii adhik pratibandhaatmak paribhasha kaa upayog karate hain, iss shart koo jodkar kii sthaan bhee ack frechet sthaan honah chaahiye. (iska matlab haye kii sthaan poorna haye aur topology seminorm key ack parivaar dwara dee jaati haye.)
nimnalikhit paribhasha kaa upayog grothendic dwara paramanu sthaanon koo paribhaashit karane key liye kiya gayaa thaa.
paribhasha 0: sootra _ 21 koo sthaaniya roop seey uttal sthalvaigyaanik sadish sthaan honey dein. phir sootra _ 21 paramanu haye yadi kissy bhee sthaaniya roop seey uttal sthaan sootra _ 23 key liye canonical vector space embedding sootra _ 24 tvss kaa ack embedding haye jisaki chhavi codoman mein ghanee haye (jahaan domen sootra _ 25 projective tensor utpaad haye aur codoman sootra _ 26 parr sabhi alag-alag nirantar dwi-rekhiya roopon kaa sthaan haye joe samvarti upasamuchchay parr samaan abhisaran kii topology key saath sampann haye). 
humm kuchh prishthabhoomi koo yaad karke shuru karate hain. ack sthaaniya uttal sthalvaigyaanik sadish sthaan sootra _ 21 mein ack topology haye jise kuchh prakaar key veerya dwara paribhaashit kiya jaataa haye. kissy bhee ardh-crum key liye, ikai gend mool kaa ack bund uttal samamit pados haye, aur isake vipareet 0 kaa koee bhee bund uttal samamit pados kissy ardh-crum kii ikai gend haye. (jatil sadish rikt sthaan key liye, sthiti "samamit" koo "santulit" dwara pratisthaapit kiya jaanaa chaahiye.) 
yadi sootra _ 28 sootra _ 29 parr ack ardh-sootra haye tou sootra _ 30 ardh-sootra sootra _ 31 kaa upayog karke sahayak maanak sthaan koo puuraa karke diye gaye banach sthaan koo darshaata haye, ack praakritik maanchitra sootra _ 32 haye (aavashyak roop seey injection naheen).
yadi sootra _ 33 ack aur ardh-roop haye, joe sootra _ 28 seey badaa haye (sootra _ 21 parr ack falan key roop mein bindu kii tarah), tou sootra _ 36 seey sootra _ 30 taka ack praakritik maanchitra haye jaise kii sootra _ 38 key roop mein pahale maanchitra kaarak yeye maanchitra hamesha nirantar hotey hain. antariksh sootra _ 21 paramanu haye jaba ack majboot sthiti hoti haye, arthaat yeye maanchitra paramanu sanchaalak hain. paramanu sanchaalak honey kii sthiti sookshm haye, aur adhik vivaran sambandhit lekh mein upalabdh hain.
paribhasha 1: ack paramanu sthaan ack sthaaniya uttal topological vector sthaan haye jaise kii kissy bhee seminorm sootra key liye humm ack badaa seminorm sootra _ 33 paa sakate hain taaki praakritik maanchitra sootra _ 42 paramanu hoe.
anaupchaarik roop seey, iska matlab haye kii jaba bhee hamein kissy seminorm kii ikai gend dee jaati haye, tou humm isake ander kissy anya seminorm kii "bahut chhoti" ikai gend paa sakate hain, yaa kii 0 key kissy bhee pados mein "bahut chhota" pados hotaa haye. sabhi seminorm sootra _ 28 key liye iss sthiti kii jaanch karna aavashyak naheen haye; yeh seminorm key ack sett key liye jaanch karane key liye paryaapt haye joe topology utpann kartaa haye, dusare shabdon mein, seminorm kaa ack sett joe topology key liye ack upa-aadhar haye.
manmaane dhang seey banach space aur paramanu operators kaa upayog karane key bajaay, humm hilbert space aur trace class operators key sandarbh mein ack paribhasha dey sakate hain, jinhen samajhana aasaan haye. 
humm kahenge kii ack ardh-sootra sootra _ 28 ack hilbert ardh-sootra haye yadi sootra _ 30 ack hilbert sthaan haye, yaa samkaksh roop seey yadi sootra _ 28 sootra _ 47 parr ack sesquilinear sakaratmak ardh-parimit roop seey aataa haye.
paribhasha 2: ack paramanu sthaan ack topological vector sthaan haye jisamen hilbert seminorm key parivaar dwara paribhaashit ack topology haye, jaise kii kissy bhee hilbert seminorm sootra key liye humm ack badaa hilbert seminorm sootra _ 33 paa sakate hain taaki sootra _ 36 seey sootra _ 30 taka kaa praakritik maanchitra trace varg hoe.
kuchh lekhak trace class operators key bajaay hilbert-schmitt operators kaa upayog karna pasand karate hain. isase bahut cumm fark padata haye, kyonki koee bhee trace class operator hilbert-schmitt hotaa haye, aur dou hilbert-schmitt operators kaa utpaad trace class kaa hotaa haye.
paribhasha 3: ack paramanu sthaan ack topological vector sthaan haye jisamen hilbert seminorm key parivaar dwara paribhaashit ack topology haye, jaise kii kissy bhee hilbert seminorm sootra key liye humm ack badaa hilbert seminorm sootra _ 33 paa sakate hain taaki sootra _ 36 seey sootra _ 30 taka kaa praakritik maanchitra hilbert-schmitt hoe.
yadi humm ack paramanu prachaalak kii avadharana kaa upayog ack manmaane sthaaniya uttal sthalvaigyaanik sadish sthaan seey ack banach sthaan taka karane key liye taiyar hain, tou humm nimnanusar chhoti paribhasha dey sakate hainh
paribhasha 4: ack paramanu sthaan sthaaniya roop seey uttal topological vector sthaan haye jaise kii kissy bhee ardh-sootra sootra key liye praakritik maanchitra sootra _ 32 paramanu haye.
paribhasha 5: ack paramanu sthaan ack sthaaniya uttal sthalvaigyaanik sadish sthaan haye jaise kii ack banach sthaan key liye koee bhee nirantar raikhik maanchitra paramanu haye.
grothendic ney nimnalikhit key samaan ack paribhasha kaa upayog kiyaah
paribhasha 6: ack paramanu sthaan ack sthaaniya roop seey uttal sthalvaigyaanik sadish sthaan sootra haye joe kissy bhee sthaaniya roop seey uttal sthalvaigyaanik sadish sthaan sootra key liye praakritik maanchitra prakshepak seey sootra 58 key injective tensor utpaad taka aur sootra 59 ack samroopata haye.
vaastav mein yeh sirf banach space formula _ 62 key liye yaa yahaan taka kii bilkul abhisari shrinkhala key ekal banach space formula _ 63 key liye bhee jaanch karane key liye paryaapt haye.
charitra varnan.
sootra _ 21 koo sthaaniya roop seey uttal sthaan maanaa jaae. taba nimnalikhit samatulya hainh
yadi sootra _ 21 ack frechet sthaan haye tou nimnalikhit samatulya hainh
paryaapt paristhitiyaan.
maan leejiye kii sootra 99 aur sootra 100 sthaaniya roop seey uttal sthaan hain aur sootra 100 paramanu haye. 
udaaharan.
yadi sootra _ 108 kissy bhee cardinality kaa ack sett haye, tou sootra _ 109 aur sootra _ 110 (utpaad topology key saath) donon paramanu sthaan hain. 
paramanu sthaan kaa ack apekshaakrit saral anant ayami udaaharan teji seey ghatate sabhi anukramon key sootra 111 kaa sthaan haye ("teji seey ghatate" kaa arth haye kii sootra 112 kissy bhee bahupad sootra _ 28 key liye bandha huaa haye). pratyek vaastavik sankhya sootra 114 key liye ack maanak sootra 115 koo paribhaashit karna sambhav haye 
sootra _ 116
yadi iss maanak mein puuraa honah sootra 117 haye tou sootra 118 seey ack praakritik maanchitra haye jaba bhee sootra 119 aur yeh paramanu haye jaba bhee sootra 120 anivarya roop seey kyonki shrinkhala sootra 121 taba bilkul abhisari haye. vishesh roop seey pratyek maanak sootra 122 key liye ack aur maanak khojana sambhav haye, sootra 123 kahein kii maanchitra sootra 124 paramanu haye. tou antariksh paramanu haye. 
guna.
paramanu sthaan kaee mayanon mein seemit-ayami sthaanon key samaan hain aur unake kaee achche guna hain.
kernell pramey.
paramanu sthaanon key adhikansh siddhant koo alekjendar grothendic dwara schwartz kernell pramey kii jaanch karate hue vikasit kiya gayaa thaa aur ise 2006 mein prakaashit kiya gayaa thaa. hamaare paas pramey kaa nimnalikhit saamaanyikaran haye. 
schwartz kernell prameyah maan leejiye kii sootra _ 21 paramanu haye, sootra _ 135 sthaaniya roop seey uttal haye, aur sootra _ 136 sootra _ 137 parr ack nirantar dwimitiya roop haye, tou sootra _ 136 sootra _ 139 key ack sthaan seey utpann hotaa haye jahan sootra _ 140 aur sootra _ 141 sootra _ 92 aur sootra _ 143 key samatulya key upayukt samvarti upasamuchchay hain, sootra _ 136 roop kaa haye, 
sootra _ 145
jahan sootra _ 145 aur sootra _ 147 aur sootra _ 148 mein seey pratyek samvarti hain. isake alawa, inn anukramon koo kramshah sootra _ 149 aur sootra _ 150 mein shoonya anukram (yaani, 0 mein abhisari) key roop mein liya jaa sakata haye.
bochner-minlos pramey.
paramanu antariksh sootra _ 58 parr ack nirantar karyatmak sootra _ 151 koo ack visheshta karyatmak sootra _ 153 kahaa jaataa haye aur kissy bhee jatil sootra _ 154 sootra _ 155 key liye. 
sootra _ 156
paramanu antariksh sootra _ 157 parr ack visheshta karyatmak diye jaane parr bochner-minlos pramey (salomon bochner aur robbert edol 'fowich minlos key baad) dohare antariksh sootra _ 159 parr ack sambandhit sambhavana maap sootra _ 158 key astitva aur vishishtata kii guaranti detaa haye.
sootra _ 160
yeh pratilom forier parivartan koo paramanu sthaanon mein vistaarit kartaa haye.
vishesh roop seey, yadi sootra _ 58 paramanu sthaan haye
sootra _ 162
jahan sootra _ 163 hilbert rikt sthaan hain, bochner-minlos pramey visheshta falan sootra _ 164 key saath ack sambhaavyata maap key astitva kii guaranti detaa haye, joe kii dohri jagah parr gausi maap kaa astitva haye. iss tarah key maap koo safed shore maap kahaa jaataa haye. jaba sootra _ 58 schwartz sthaan haye, tou sambandhit yaadrichhik tatva ack yaadrichhik vitaran haye.
majboot paramanu sthaan.
ack majboot paramanu sthaan ack sthaaniya uttal topological vector sthaan haye jaise kii kissy bhee seminorm sootra key liye ack badaa seminorm sootra _ 33 maujood haye taaki praakritik maanchitra sootra _ 42 ack majboot paramanu hoe.