
anhu antariksham

ganita saastramlo, nuclear spacelu anevi topological vector spacelu, veetini parimita dimensional euclidean spacela saadhaaraneekaranagaa chudavachu mariyu vaati kavalsina lakshanaalalo chaalaa vaatini panchukovachhu. ayithe anhu pradaesaalu hilbert pradaesaalaku chaalaa bhinnamgaa untaayi, idhi parimita dimensional euclidean pradaesaala yokka maroka saadhaaraneekarana. vaatini alegzander grothendic parichayam chesaaru.
anhu pradaesaalapai sthalaakrutini seminarmla kutumbam dwara nirvachinchavacchu, dheeni unit bantulu parimaanamlo vaegamgaa taggutaayi. muulakaalu edho oka konamlo "mruduvaina" vector pradaesaalu anhu pradaesaalugaa untaayi; anhu pradaesaaniki oka vilakshanhamaina udaaharana compact manifold meeda mruduvaina vidhula samiti. anni parimita-dimensional vector khaaliilu nuclear. parimita-dimensional khaaliilu tappa, anugaa unde banach khaaliilu laevu. aacharanalo dheeniki oka vidhamaina paraspara charya tarachugaa nijam avtundi "sahajamgaa sambhavinche" sthalavignaana vector sthalam oka banach space ayithe, adhi anugaa undataaniki manchi avakaasam undhi.
asalu preranah schwartz kernell siddhaantam.
squartz kernell siddhaantaanni parisoedhistunnappudu alegzander grothendic anhu pradaesaala siddhaantaanni chaalaa varaku abhivruddhi chesi, 2006loe prachurinchaaru. ippudu manam eee prerananu vivaristaamu. 
edainaa open sub setla formula _ 1 mariyu formula _ 2 kosam canonical mapp formula _ 3 anedi tvss yokka isomorphism (ikkada formula _ 4 parimita sub setlapai ekareeti convergence yokka topologyni kaligi untundi) mariyu inca, eee rendu khaaliilu kananikalga tvs-isomorphic tuu formula _ 5 (ikkada formula _ 6 nuclear kabatti, eee tensor utpatti ekakaalamlo injective tensor utpatti mariyu projective tensor utpatti). 
sankshiptamgaa, schwartz kernell siddhaantam ilaa perkondih
suutram _ 7
ikkada eee tv-isomorphisms annee canonical. 
evarainaa space formula _ 8 nuu formula _ 9 thoo bharti chesi (idhi daani swantha balamaina dual spaceku samaruupamgaa unde reflexive space) mariyu formula _ 10 nuu eee formula _ 9 space yokka dualtho bharti chesthe eee phalitam tappu. 
pampinheelu mariyu pareeksha vidhula sthalaaniki intha manchi phalitam enduku untundi, conny hilbert space formula _ 9 (idhi saadhaaranamgaa "nicest" tvs laloe okatigaa pariganinchabadutundi) kosam kaadu? 
eee prasna grothendic anhu pradaesaalu, anhu pataalu mariyu injective tensor utpattini kanugonadaaniki daariteesindi.
jyaamiti nundi preranalu.
maroka prerepinche udaaharanala samiti neerugaa jyaamiti mariyu mruduvaina manifold siddhaantam nundi vastundi. mruduvaina manifold suutram _ 13 mariyu sthaanikamgaa kumbhakara hasdorff topological vector space ichinatlayithe, appudu anhu pradaesaala yokka eee krindi isomorphisms unnaayi.
suutram _ 16 kosam praamaanika tensor utpattulanu vector spacega upayoginchadam dwara, function formula _ 17nii suutram _ 19 kosam suutram _ 18 gaa vyakteekarinchalemu, idhi suutram _ 20 yokka setlanu khachitamgaa cherchadam undani niroopinche udaaharananu istundi.
nirvachanam.
eee vibhagam anhu sthalam yokka konni saadhaarana nirvachanaalanu jabita chestundi. diguva nirvachanaalannii samanamga untaayi. kontamandi rachayitalu anhu sthalam yokka marinta nirbandhaatmaka nirvachananni upayogistaarani gamaninchandi, sthalam kooda oka frechet space ayi undaali anae sharatu jodinchadam dwara. (dheeni ardham sthalam puurtayindani mariyu topologyni seminarmla kutumbam istundi.)
anhu pradaesaalanu nirvachinchadaaniki grothendic eee krindi nirvachananni upayoginchaaru.
nirvachanam 0: suutram _ 21 sthaanikamgaa kumbhakara sthalavignaana vector spacega undanivvandi. edainaa sthaanikamgaa kumbhakara antariksha suutram kosam canonical vector space embedding formula _ 24 anedi codominelo dattamgaa unde tvss yokka embedding ayithe suutram _ 21 nuclear avutundi (ikkada domine formula _ 25 anedi projective tensor utpatti mariyu codomine anedi suutram _ 26 paie anni vidigaa nirantara dwiparshwa roopaala sthalam, idhi samaanamaina upasamitulapai ekareeti kalayika yokka topologytho nindi untundi). 
kontha nepathyaanni gurtuchesukovadam dwara praarambhistaamu. sthaanikamgaa kumbhakara sthalavignaana vector space formula _ 21 oka sthalavignaanaanni kaligi undhi, idhi konni seminarmla kutumbam dwara nirvachinchabadutundi. edainaa seminarm kosam, unit banthi muulam yokka closed convex cemetric parisaraalu, mariyu dheeniki viruddhamgaa 0 yokka edainaa closed convex cemetric parisaraalu konni seminarm yokka unit banthi. (sanklishta vector pradaesaala kosam, "cemetric" anae sharatu "balanced" thoo bharti cheyali.) 
suutram _ 28 suutram _ 29 paie oka seminarm ayithe, suutram _ 30 anedi seminarm suutram _ 31nii upayoginchi sahayaka praamaanika sthalaanni puurti cheyadam dwara icchina bananach sthalaanni suuchistumdi, sahaja mapp suutram _ 32 undhi (tappanisarigaa injective kaadu).
suutram _ 33 anedi suutram _ 28 kante peddadi ayithe (suutram _ 21 paie functionga binduvulugaa), suutram _ 36 nundi suutram _ 30 varaku sahaja patam untundi, tadwara suutram _ 38 vanti modati pata karakalu ellappuduu nirantaramgaa untaayi. balamaina paristhiti unnappudu antariksha suutram _ 21 anhu, antey eee pataalu anhu operatorlu. anhu operator anae paristhiti suukshmamgaa untundi, mariyu marinni vivaralu sambandhita vyaasamlo andubaatulo unnaayi.
nirvachanam 1: nuclear space anedi sthaanikamgaa kumbhakara topological vector space, antey edainaa seminarm formula _ 28 kosam manam pedda seminarm formula _ 33 nii kanugonavacchu, tadwara sahaja mapp formula _ 42 nuclear avutundi.
anadhikarikamga, dheeni ardham manaku edainaa seminarm yokka unit banthi ichinappudalla, daani lopala maroka seminarm yokka "chaalaa chinna" unit bantini kanugonavacchu, ledhaa 0 yokka edainaa parisaraalu "chaalaa chinna" porugu praantaanni kaligi untaayi. anni seminarmla formula _ 28 kosam eee paristhitini tanikhi cheyavalasina avasaram ledhu; topologyni utpatti chese seminarmla samiti kosam tanikhi cheyadam saripotundi, maroe maatalo cheppalante, topologyki upa aadhaaramaina seminarmla samiti.
ekapaksha banach pradaesaalu mariyu nuclear operatorlanu upayoginchadaaniki badulugaa, manam hilbert pradaesaalu mariyu trace class operatorla paramgaa oka nirvachananni ivvavacchu, ivi ardham chesukovadaniki sulabham. 
suutram _ 30 anedi hilbert space ayithe, ledhaa suutram _ 28 suutram _ 47 paie sesquilinear positive semidifinite roopam nundi vasthe, daaniki samanamga seminarm formula _ 28 anedi hilbert seminarm anie chebutamu.
nirvachanam 2: nuclear space anedi hilbert seminarmla kutumbanche nirvachinchabadina topologytho kuudina topological vector space, antey edainaa hilbert seminarm formula _ 28 kosam manam pedda hilbert seminarm formula _ 33nii kanugonavacchu, tadwara formula _ 36 nundi formula _ 30 varaku sahaja mapp trace class avutundi.
kontamandi rachayitalu trace class operatorlanu kakunda hilbert-schmid operatorlanu upayoginchadaaniki ishtapadataaru. idhi peddagaa thedaa chuupadu, endukante edainaa trace class operator hilbert-schmid, mariyu iddaru hilbert-schmid operatorla utpatti trace class.
nirvachanam 3: nuclear space anedi hilbert seminarmla kutumbanche nirvachinchabadina topologytho kuudina topological vector space, antey edainaa hilbert seminarm formula _ 28 kosam manam pedda hilbert seminarm formula _ 33nii kanugonavacchu, tadwara formula _ 36 nundi formula _ 30 varaku sahaja mapp hilbert-schmid.
manamu nuclear operator yokka bhavananu ekapakshamgaa sthaanikamgaa kumbhakara sthalavignaana vector sthalam nundi banach sthalam varaku upayoginchadaaniki siddhamgaa unte, manamu eee krindi vidhamgaa chinna nirvachanaalanu ivvavachuh
nirvachanam 4: nuclear space anedi sthaanikamgaa kumbhakara topological vector space, antey edainaa seminarm formula _ 28 kosam formula _ 32 nundi sahaja mapp nuclear.
nirvachanam 5: nuclear space anedi sthaanikamgaa kumbhakara topological vector space, antey banach space kii edainaa nirantara sarala mapp nuclear.
grothendic eee krindi nirvachananni pooli oka nirvachananni upayoginchaaduh
nirvachanam 6: nuclear space anedi sthaanikamgaa kumbhakara topological vector space formula _ 58, antey edainaa sthaanikamgaa kumbhakara topological vector space formula _ 59 kosam projective nundi suudi mandam tensor utpatti suutram _ 58 mariyu formula _ 59 varaku sahaja mapp oka isomorphism.
vaastavaaniki dheenini kevalam banach space formula _ 62 kosam ledhaa puurtigaa convergent syries yokka single banach space formula _ 63 kosam kooda tanikhi chesthe saripotundi.
lakshanaalanu.
suutram _ 21 sthaanikamgaa kumbhakara sthalamgaa undanivvandi. appudu eee krindivi samanamga untayih
suutram _ 21 oka frechet space ayithe, eee krindivi samanamga untayih
taginanta paristhitulu.
suutram _ 99 mariyu suutram _ 100 anevi nuclear suutram _ 100 thoo sthaanikamgaa kumbhakara sthalam anie anukundam. 
udaaharanalu.
formula _ 108 edainaa cardinality samiti ayithe, formula _ 109 mariyu formula _ 110 (utpatti topologytho) rendoo nuclear spacelu. 
anhu sthaanaaniki saapekshamgaa saralamaina anantamaina dimensional udaaharana vaegamgaa taggutunna anni sequencela formula _ 111 ("vaegamgaa taggutunna" antey edainaa bahupadi formula _ 28kii formula _ 112 kattubadi untundi). prati vaastava sankhya suutram _ 114kii oka praamaanika suutram _ 115nii nirvachinchadam saadhyamavutundi 
suutram _ 116
eee pramaanamloo puurti suutram 117 ayithe suutram 119 nundi sahaja patam untundi mariyu suutram 120 ayinappudalla idhi nuclear avutundi endukante syries suutram 121 appudu puurtigaa convergent avutundi. mukhyamgaa prati praamaanika suutram _ 122kii idhi maroka pramaanaanni kanugonadam saadhyamavutundi, suutram _ 123 anie cheppandi, antey mapp suutram _ 124 anhu. kabatti sthalam anhu. 
aastulu.
anhu pradaesaalu aneka vidhaluga parimita-dimensional pradaesaalatoe samanamga untaayi mariyu vaati manchi lakshanaalanu kaligi untaayi.
kernell siddhaantam.
squartz kernell siddhaantaanni parisoedhistunnappudu alegzander grothendic anhu pradaesaala siddhaantaanni chaalaa varaku abhivruddhi chesi, 2006loe prachurinchaaru. manaku siddhaantam yokka eee krindi saadhaaraneekarana undhi. 
squartz kernell siddhaantam suutram _ 21 anhu, suutram _ 135 sthaanikamgaa kumbhakaramga, mariyu suutram _ 136 suutram _ 137 paie nirantara dwiparshwa roopam anie anukundam, appudu suutram _ 136 suutram _ 139 yokka sthalam nundi udbhavistundi, ikkada suutram _ 140 mariyu suutram _ 141 suutram _ 92 mariyu suutram _ 143 yokka tagina samaanamaina upasamitulu, suutram _ 136 roopamlo untundi, 
suutram _ 145
ikkada suutram _ 145 mariyu suutram _ 147 mariyu suutram _ 148 prati okkati samanamga untaayi. inca, eee shrenulanu varusaga _ 149 suutram mariyu _ 150 suutramloe shoonya sraenulugaa (antey, 0kii convergent) teesukovachhu.
boachner-minlos siddhaantam.
anhu antariksha suutram _ 58 paie nirantara kriyaatmaka suutram _ 153 suutram _ 153 ayithe mariyu edainaa sanklishta suutram _ 154 suutram _ 155 kosam lakshana kriyaatmaka suutram anie pilustaaru. 
suutram _ 156
nuclear space formula _ 157 paie oka lakshana kriyaatmakatanu icchinappudu, boachner-minlos siddhaantam (saloman boachner mariyu robbert adol 'fovich minlos taruvaata) dual space formula _ 159 paie sambandhita sambhaavyata kolata formula _ 158 yokka uniki mariyu pratyaekatanu haamii istundi.
suutram _ 160
idhi inverse forier parivartananu nuclear spacelaku vistaristundi.
mukhyamgaa, suutram _ 58 anhu sthalam ayithe
suutram _ 162
suutram _ 163 anedi hilbert khaaliilu, boachner-minlos siddhaantam vilakshanhamaina function suutram _ 164 thoo sambhaavyata kolata unikiki haamii istundi, antey dwandwa sthalamlo gassian kolata uniki. atuvanti kolatanu white nais kolata antaaru. suutram _ 58 anedi schwartz space ayinappudu, sambandhita yaadruchchika moolakam yaadruchchika pampinhee.
balamaina anhu pradaesaalu.
balamaina anhu sthalam anedi sthaanikamgaa kumbhakara sthalavignaana vector sthalam, antey edainaa seminarm suutram _ 28 kosam pedda seminarm suutram _ 33 untundi, tadwara sahaja pata suutram _ 42 balamaina anhu.

