
amoort winer jaaga

amoort winer jaagechi sankalpana hii eka ganiteey rachana aahe jee leonard gross yanni anant-aayami jaganvaril gossian mojamapachi rachana samajoon ghenyasathi vikasit kelee aahe. bandhkam cameron-martin jaagene bajavalelya moolbhut bhoomikevar bharr dete. shastriy winer space heey eka aadarsh udaharan aahe.
gossian mojmapansathichya sanrachana prameyat asey mhatale aahe qee serve gossian mojmap amoort winer space bandhkamadware darshavile jaau shakatat.
prerana.
sootra _ 1 hii eka vastavik hilbert jaaga asoo dya, jee anant aayami aanee vegale karnyayogya aahe asey gruheet dharale jaate. bhautikshastra sahityat, ekhadyala varanvar yaa swarupachya samakalanancha samna karava lagato 
jethe sootra _ 3 haah samanyikaran sthirank manla jaato aanee jethe sootra _ 4 heey sootra _ 5 vareel astitvat nasalele lebsg mojmap manale jaate. asey samakalan udbhavtat, visheshtah, quantum feeled thearychya euclidean paath-integral formulation sandarbhat. ganiteey staravar, asha avibhajya bhagacha arth mool hilbert space sootra _ 5 vareel mojamapachya viruddha ekikaran assa lavala jaau shakat nahee. dusarikade, samaja sootra _ 7 hii eka banach space aahe jyamadhye ghanata upkshetra mhanoon sootra _ 5 aahe. jarr sootra _ 7 heey sootra _ 5 peksha "purese mothe" asel, tarr vareel avibhajya sootra _ 7 vareel changlya prakare paribhashit (gossian) mojamapachya viruddha ekikaran mhanoon arth lavala jaau shakato. asha paristhitit, jodee sootra _ 12 laa amoort winer space mhanoon sambodhale jaate.
shastriy winer space heey aadarsh udaharan aahe, jyamadhye sootra _ 5 heey vastavik-moolya functionsche hilbert space aahe, sootra _ 15 madheel sutravar sootra _ 14, sootra _ 16 madhye eka vyutpann aanee samadhankarak sootra _ 17 aahe, jyadvare manak dile jaate
asha paristhitit, sootra _ 7 heey sarvoch mankansah sootra _ 15 vareel nirantar karyanchi banach space mhanoon ghetale jaau shakate. yaa prakaranat, sootra _ 7 vareel mojmap heey moodpasoon suru honanya braunian gatiche varnan karnare winer mojmap aahe. mool subspace sootra _ 22 laa cameron-martin space mhanatat, jeey winer mojamapachya sandarbhat mojamapachya shoonyacha sanch tayar karate.
aadhichya udaharanacha arth assa aahe qee, jyane dilelya winer mojamapasathi aamchyakade eka "aupcharik" abhivyakti aahe
sootra _ 23
jaree hii aupcharik abhivyakti "suchavate" qee winer mojmap jyaa marganchya jaaget _ 24 sutrasathi asave, pratyakshat asey nahee. (braunian marg sambhavyatesah kuthehi bhinn karnyayogya naheet asey olakhale jaate.) 
grossche amoort winer space bandhkam shastriy winer spesasathichi paristhiti sarbhut karate aanee sootra _ 7 varr astitvat asalelya gossian mojamapasathi aavashyak aanee pureshi (jarr kadhikadhi tapasane katheen asel tarr) aut pradan karate. jaree gossian mojmap sootra _ 26 sootra _ 5 aivaji sootra _ 7 varr rahate, taree sootra _ 7 aivaji sootra _ 5 chee bhoomiti aahe jee sootra _ 26 chya gunadharmanvar niyantran thevate. sthool swatah mhanato (aaplya sanketankanusar), "tathapi, heey fakt i chya karyane spasht jhaale.eae. vastavik hilbert jagechya samanya vitaranashi nigadit segal, hilbert space formula _ 5 chee bhoomika kharokharach madhyavarti hotee aanee sootra _ 7 vareel vishleshanacha vichar karata, formula _ 7 chee bhoomika swatach anek cameron aanee martinchya prameyansathi sahayak hotee aanee kahee prakarananmadhye agadi anavashyak dekhil hotee."sthoolchya amoort winer antaral bandhkamachya aakarshak vaishishtyanpaiki eka mhanje teey sootra 5 haah prarambhik bindu mhanoon ghete aanee sootra 7 laa sahayak vastoo mhanoon manate.
yaa vibhagat aadhi disana sootra _ 26 sathichya aupcharik abhivyakti jaree purnapane aupcharik, bhautikshastra-shailichya abhivyakti asalya taree, tyaa sootra _ 26 chey gunadharm samajoon ghenyas madat karnyasathi khoop upayukt aahet. visheshtah, ekhadi vyakti (barobar!) sootra 41 sathee, sootra 40 chya sapeksh anuvadit mojmap sootra 39 chya ghanatesathi sootra. (cameron-martin pramey pahaa.)
ganiteey varnan.
cylinderche mojmap chaloo qaraa.
sootra 5 heey vastavik sankhyanvar paribhashit kelele hilbert space asoo dya, jeey anant aayami aanee vegale karata yenyajoge aahe asey gruheet dharale jaate. sootra 5 madheel eka cylinder sanch haah sootra 5 vareel rekheey karyanchya maryadit sankalanachya mulyanchya drushtine paribhashit kelela sanch aahe. visheshtah, samaja sootra 45 heey sootra 5 vareel satat rekheey karye aahet aanee sootra 47 heey sootra 48 madheel borel sanch aahe. mug aapan sanch vicharat gheu shakato.
sootra _ 49
yaa prakarchya konatyahi sanchala nall sanch mhanatat. serve nall sanchancha sangrah sootra 5 madheel sanchancha bijganit tayar karato parantu too sootra 51-bijganit nahee.
nalake sanchavar "mojmap" paribhashit karnyacha eka naisargik marg aahe, joou khalilpramane aahe. rez prameyanusar, rekheey karyatmak sootra _ 52 heey sootra _ 5 madhye vahak sootra _ 53 saha antargat utpadan mhanoon dile jaate. graam-schmitt prakriyechya prakashat, sootra _ 53 heey orthonormal aahet asey gruheet dharane nirupadravi aahe. asha paristhitit, aapan vareel paribhashit cylinder sanch sootra _ 56shhee sootra _ 58 vareel pramanit gossian mojamapachya sandarbhat sootra _ 47 chey mojmap jodu shakato. mhanje, aapan paribhashit karato.
sootra _ 59
jethe sootra _ 60 heey sootra _ 48 vareel manak lebsgue mojmap aahe. sootra _ 48 vareel manak gossian mojamapachya utpadan sanrachanemule, sootra _ 26 changlya prakare paribhashit aahe heey darshavine katheen nahee. mhanjech, jaree ekach sanch sootra _ 56 heey ekapeksha jaast prakare sanchit cylinder mhanoon darshavile jaau shakate, taree sootra _ 65 chey moolya nehmich samaan asate.
upay astitvat nahee.
sanch karyatmak sootra _ 26 laa sootra _ 5 vareel manak gossian cylinder sanch mojmap mhanatat. asey gruheet dharoon (jase aapan karato) qee sootra _ 5 heey anant aayami aahe, sootra _ 26 "sootra _ 5 madheel cylinder sanchanchya sankalanadvare tayar kelelya sootra _ 51-bijganitavaril gananiy yogatmak mojamapasathi vistarat nahee. sootra _ 72 varr dilelya manak gossian mojamapachya vartanacha vichar karoon ekhadi adchan samajoo shakate.
sootra _ 73
yaa mojamapachya sandarbhat varg manakache apekshit moolya prathamik gossian avibhajya mhanoon mojale jaate
sootra _ 74
mhanjech, sootra _ 48 vareel pramanit gossian mojmapasarkhe yadruchchikpane nivadalele vectorchya utpattipasoonche vaishishtyapurn antar sootra _ 76 aahe kaaran sootra _ 77 anantakade valate, heey vaishishtyapurn antar anantakade valate, jeey soochit karate qee sootra _ 5 varr konatehi suspasht "manak gossian" mojmap nahee. (moodapasoonche vaishishtyapurn antar anant asel, jenekarun mojmap pratyakshat avkash sootra _ 5 varr tikanar nahee.)
upay astitvat.
aata samaja qee sootra _ 7 hii eka vegale karata yenyajogi banach jaaga aahe aanee teey sootra _ 81 haah eka injective satat resheey nakasha aahe jyachi pratima sootra _ 7 madhye daat aahe. nantar sootra _ 5 laa sootra _ 7 madheel tyachya pratimesah olakhane nirupadravi (aanee soyiche) aahe aanee asha prakare sootra _ 5 laa sootra _ 7 chaa ghan upasanch mhanoon vicharat ghya. tyanantar aapan sootra _ 7 varr eka cylinder sanch mojmap tayar karoo shakato, jyamadhye cylinder sanch sootra _ 88 chey mojmap heey sootra _ 89 chey purvi paribhashit kelele cylinder sanch mojmap aahe, jeey sootra _ 5 madhye sanchit eka cylinder aahe.
amoort winer space construction kalpana ashee aahe qee jarr sootra _ 7 heey sootra _ 5 peksha purese mothe asel, tarr sootra _ 5 vareel cylinder seet mojamapachya ulat, sootra _ 7 vareel cylinder seet mojamapache mojmap, tayar kelelya sootra _ 51-bijganitavaril mojmapyogya yogatmak mojmapasarkhe hoil. ekoon mool kagad heey tase honyasathi sootra _ 7 varr aavashyak aanee pureshi aut deto. sootra _ 7 vareel mojamapalela gossian mojmap mhanatat aanee upkshetra sootra _ 22 laa cameron-martin space mhanatat. heey adhorekhit karane mahatvache aahe qee sootra 5 heey sootra 7 chya aat shoonyacha sanch tayar karate, jyavar jore detana qee gossian mojmap heey keval sootra 7 varr rahate aanee sootra 5 varr nahee.
yaa sampoorn charchecha parinam assa aahe qee prerana vibhagat varnan kelelya prakarchya gausian samakalananna kathor ganiteey arth aahe, parantu teey tyaa jaaget rahat naheet jyache manak aupcharik abhivyaktichya ghatankat aadhalate. tyaeivaji, teey kahee mothya jaaget rahatat.
bandhkamachi sarvatrikta.
amoort winer antaral bandhkam hii keval gossian upay tayar karanyachi eka paddhat nahee. tyaeivaji, anant-aayami banach jaaget 'pratyek' gausian mojmap asha prakare hote. (gossian mojmapansathi sanrachana pramey pahaa.) mhanje, anant-aayami, vegale karata yenyajogi banach jaaga (sootra _ 104 varr) varr eka gausian mojmap sootra _ 26 dile tarr, ekhada cameron-martin subspace sootra _ 22 olakhoo shakato, jyaa kshani jodee sootra _ 12 eka amoort winer jaaga banate aanee sootra _ 26 heey sambandhit gausian mojmap aahe.
udaharaneh shastriy winer jaaga.
amoort winer jaageche aadarsh udaharan mhanje satatchya marganchi jaaga aanee tyala shastriy winer jaaga mhanoon olakhale jaate. hii amoort winer jaaga aahe jyamadhye sootra _ 5 dware dile jaate
sootra _ 117
dware dilelya antargat utpadanasah
sootra _ 118
aanee sootra _ 7 heey sootra _ 15 gunile sootra _ 58 chya satatyapurn nakashanchi jaaga aahe, jee 0 pasoon suru hote, eksaman nikashasah. yaa prakaranat, gossian mojamapache sootra _ 26 heey winer mojmap aahe, jeey moodpasoon suru houn _ 58 yaa sutrat braunian gatiche varnan karate. 
sootra _ 5 heey sootra _ 26 chya sandarbhat mojamapacha shoonyacha sanch tayar karato, yaa prakaranat samanya braunian margachi khadbaditpana pratibimbit hote, jee kuthehi bhinn nahee asey dnyat aahe. heey sootra _ 5 madheel marganchya gruheet bhinnatechya viruddha aahe.