
banach beejaganitam

ganithashaasthrathil, prathyekichu pravarthanaparamaaya vishakalanathil, stephaan banaachinte peril ariyappedunna oru banach beejaganitam, yadhaartha allengil sankeernna samkhyakalkku (allengil oru archemedian ithara samboornna normed feeldinu mukalilulla) mukalilulla oru associative beejaganita soothravaakyamaanu, athu ore samayam oru banack space koodiyaanu, athaayath, maanadandam prerippikkunna metrik poortthiyaakkunna oru normed space. maanadandam thripthippeduthendathundu
soothravaakyam _ 2
gunana pravartthanam thudarchayaayi nadakkunnuvennu ithu urappaakkunnu.
gunanathinu oru aidantitti ghadakam formula _ 3 aanenkil oru banach beejaganithathe "yoonittal" ennum gunanam commutative aanenkil "commutative" ennum vilikkunnu.
eathenkilum banach beejaganita soothravaakyam _ 1 (athinu oru aidantitti ghadakam undengilum illengilum) oru ekeekritha banach beejaganita soothravaakyathil isometric aayi ulcherkkaan kazhiyum, angane formula _ 5-nte oru clossed ideal roopappeduthaam. palappozhum parigananayilulla beejaganitam ekamaanennu oraal "oru munganana" anumanikkunnu kaaranam soothravaakyam _ 5 pariganichu phalam yadhaartha beejaganithathil prayogikkunnathiloode oraalkk sidhaanthathinte bhooribhaagavum vikasippikkaan kazhiyum. ennirunnaalum, ellah samayathum ithu anganeyalla. udaaharanathinu, aidantitti illaathe oru banach beejaganithatthile ellah trikonamithika pravartthanangalum nirvachikkaan kazhiyilla.
yadhaartha banach aaljibrakalude sidhaantham sankeernamaaya banach aaljibrakalude sidhaanthathil ninnu valare vyathyasthamayirikkum. udaaharanathinu, oru nissara sankeernamaaya banach beejaganithatthile oru moolakatthinte spectrum orikkalum shoonyamaayirikkilla, athesamayam oru yadhaartha banach beejaganithathil athu chila moolakangalkk shoonyamaayirikkaam.
banachu aaljebrakale formula _ 8-aadik nambarukalude feeldukalilum nirvachikkaam. ithu formula _ 8-edic vishakalanathinte bhagamaanu.
udaaharanagal.
ananthathayil aprathyakshamaakunna praadeshikamaayi othukkamulla (housedorff) sthalathu (sankeernna-moolyamulla) thudarchayaaya pravartthanangalude idamaaya formula _ 10 aanu banach beejaganithatthinte prototypic udaaharanam. formula 12 othukkamullathaanenkil maathram formula 10 ekathaanamaanu. sankeernamaaya samyojanam oru involsion aayathinal, formula _ 10 yadhaarthathil oru sii *-aljibrayaanu. kooduthal pothuve, oro sii *-algibra nirvachanam anusarichu oru banach aljibrayaanu.
swathukkal.
power seerees vazhi nirvachikkappedunna niravadhi praadhamika pravartthanangal eathenkilum yoonittal banach beejaganithathil nirvachikkappedaam; udaaharanangalil exponential function trikonamithira pravartthanangalum, kooduthal saadhaaranayaayi eathu muzhuvan pravartthanavum ulppedunnu. (prathyekichum, amoortha soochika grouppukale nirvachikkaan exponential mapp upayogikkaam.) jyaamitheeya shreniyude soothravaakyam pothuvaayi yoonittal banack aaljibrakalil saadhuthayullathaanu. oru banach beejaganithatthinte rund sanjaara gadakangalkkum binomial sidhaantham baadhakamaan.
eathoru yoonittal banach beejaganithathileyum invertible gadakangalude koottam oru ooppan settaanu, eee settile inversion ooppareshan thudarchayaayirikkum (athinaal oru homeomorphisamaanu), athinaal ithu gunanathinu keezhil oru topological group roopeekarikkunnu.
oru banach beejaganithathinu yoonit formula 34 undengil formula 35 oru commutator aakaan kazhiyilla; athaayath, eathenkilum formula 37-nuu formula 36, kaaranam formula 38-num formula 39-num formula 40 ozhike ore spectrum undu.
melpparanja udaaharanangalil nalkiyirikkunna functionukalude vividha aaljibrakalkk riyal polulla aaljibrakalude staanderd udaaharanangalil ninnu valare vyathyasthamaaya gunangalundu. udaaharanathina
spectral sidhaantham.
sankeernamaaya feeldile yoonittal banack aaljibrakal spectral sidhaantham vikasippikkunnathin oru pothu krameekaranam nalkunnu. formula 49 kondu soochippikkunna oru moolaka formulayude "spectrum", aaa sankeernamaaya scaler formula 50 ulkkollunnu, athaayath formula 47 ll formula 51 invertible allah, eathenkilum moolaka formula 53 nte spectrum formula 54 leeii clossed diskinte oru clossed upavibhagamaanu, radius formula 55 umm sentar formula 56 umm ullathinaal kompaakt aanu. koodaathe, oru moolaka soothravaakyathinte spectrum formula _ 53 shoonyamallaathathum spectral radius formula thripthippeduthunnu
formula _ 59
formula _ 48 nalkumbol, holomorphic functional calculus formula _ 63 nte ayalpakkathulla eathu function formulakkum formula _ 61 nirvachikkaan anuvadikkunnu, koodaathe spectral mapping sidhaantham ingane parayunnu
soothravaakyam _ 64
baanaach beejaganita soothravaakyam _ 1 ennathu oru sankeernna baanaach space soothravaakyathil (udaaharanathinu, square maatriksukalude beejaganitam) bounded leanier oopparettarmaarude beejaganita soothravaakyam _ 66 aakumbol, formula _ 1 leeii spektrathekkurichu ashayam oopparettar sidhaanthathile saadhaarana ashayavumaayi poruthappedunnu. formula _ 69-nuu (compact hosdorff space formula _ 12 upayogichu), oraal kaanunnu
soothravaakyam _ 71
oru sii *-aljibrayude oru saadhaarana moolaka soothravaakyam _ 53-nte maanadandam athinte spectral aaram anusarichaanu. ithu saadhaarana oopparettarmaarkku samanamaaya oru vasthuthaye saamaanyavalkkarikkunnu.
formula _ 1 oru sankeernamaaya ekeekritha banach beejaganithamayirikkatte, athil poojyamallaatha ellah moolaka soothravaakyangalum _ 53 vipareethamaanu (oru division beejaganitam). oro soothravaakyathinum _ 75 enna soothravaakyam undu, athaayath
formula _ 77 vipareethamalla (kaaranam formula _ 78-nte spectrum shoonyamalla) athinaal formula _ 79 eee beejaganita formula _ 1 swaabhaavikamaayum formula _ 54 (gelfand-masoor sidhaanthathinte sankeernna cases)-nuu isomorphic aanu.
aadarshangalum kathaapaathrangalum.
formula _ 1 oru ekeekritha "commutative" banach beejaganithamayirikkatte, kaaranam formula _ 1 unitulla oru commutative ring aayathinal, formula _ 1-nte oro noonn-invertable gadakavum formula _ 47-nte eathenkilum maximal idealil ulppedunnu, kaaranam formula _ 1-leeii oru maximal ideal formula _ 87 adanjirikkunnu, formula _ 89 oru banach aljibrayaanu, athu oru feeldaanu, koodaathe jelfand-masoor sidhaanthathil ninnu formula _ 1-nte ellah maximal idealukaludeyum koottatthinum formula _ 1 muthal formula _ 83 vareyulla ellah poojyamallaatha homomorphisangalude settu formula _ 91-num idayil oru baijakshan undennu parayunnu.
oru akshara formula _ 96 ennathu formula _ 1-leeii oru leanier functionalaanu, athu ore samayam gunanavum formula _ 98 umm aanu, koodaathe formula _ 99-neey thripthippeduthunnu, oro aksharavum formula _ 1 muthal formula _ 101 vare yaanthrikamaayi thudarnnupokunnu, kaaranam oru pratheekatthinte kernel oru maximal idealaanu, athu closs cheythirikkunnu. koodaathe, oru kathaapaathrathinte maanadandam (athaayath, oopparettar maanadandam) onnaanu. formula _ 1 (athaayath, formula _ 103-nte durbalamaaya topology indues cheytha topology)-ll pooyint vais kanverjansinte topology sajjeekarichirikkunna akshara idam, formula _ 104 oru hosdorff compact space aanu.
eathenkilum formulakku _ 48
soothravaakyam _ 106
formula _ 107 ennathu formula _ 53-nte jelfand praathinidyamaanu, thaazhepparayunna reethiyil nirvachichirikkunnu formula _ 107 ennathu formula _ 91 muthal formula _ 54 vareyulla thudarchayaaya pravarthanamaanu, formula _ 112 nalkia formula _ 113-nte spectrum, compact space formula _ 115-leeii sankeernamaaya thudarchayaaya pravartthanangalude beejaganita formula _ 114-nte gadakamaayi spectramaanu,
soothravaakyam _ 116
oru beejaganitam enna nilayil, oru yoonittal commutative banach beejaganitam athinte jelfand praathinidhyathinu nissaramaaya kernel undengil maathrame ardha lalithamaakoo (athaayath, athinte jacobson radical poojyamaanu). atharamoru beejaganithatthinte oru pradhana udaaharanam oru commutative sii *-beejaganithamaanu. vaasthavathil, formula _ 1 oru commutative yoonittal c *-aljibra aayirikkumbol, jelfand praathinidhyam formula _ 1 num formula _ 119 num edayilulla oru isometric *-isomorphisamaanu.
banach *-algebras.
oru banach *-aljibra formula _ 1 ennathu sankeernna samkhyakalude feeldil oru banach aljibrayaanu, oppam inipparayunna gunangalulla oru mapp formula _ 121 umm undu
mattoru vidhathil paranjaal, oru banach *-beejaganitam ennathu formula _ 54 nuu mukalilulla oru banach beejaganithamaanu, athu oru *-beejaganitam koodiyaanu.
mikka swaabhaavika udaaharanangalilum, involsion isometric aanennum oraalkk undu, athaayath, 
formula _ 134
chila rachayithakkal eee isometric property oru banach *-aljibrayude nirvachanathil ulppeduthunnu.
oru banach *-aljibra thripthikaramaaya soothravaakyam _ 135 oru sii *-aljibrayaanu.

