seminome

ganithashaasthrathil, prathyekichu pravarthanaparamaaya vishakalanathil, oru seminome ennathu positteev definate aayirikkendathilla. seminormukal convex settukalumaayi adutha bandhamullavayaanu oro seminorm chila agiranam cheiyunna diskinte minkovsky pravarthanamaanu, neremarichu, atharam eathenkilum settinte minkovsky pravartthanam oru seminormaanu.
oru topological vekttar space athinte topology seminormukalude oru kudumbathaal prachodippikkappettaal maathrame praadeshikamaayi convex aakoo.
nirvachanam.
formula _ 1 ennathu yadhaartha samkhyakalude formula _ 2 allengil sankeernna samkhyakalude formula _ 3 nuu mukalilulla oru vekttar space ayirikkatte. 
thaazhepparayunna rund vyavasthakal thripthippeduthukayaanel oru yadhaartha moolyamulla function formula _ 4 neey a ennu vilikkunnu
eee rund vyavasthakalum soochippikkunnathu formula _ 10 umm oro seminome formula _ 11 num inipparayunna gunamunda
chila rachayithakkal "seminome" enna nirvachanatthinte bhaagamaayi (chilappol "maanadandam") noonn-negativittiye ulppeduthunnu, ennirunnaalum ithu mattu rund gunangalil ninnu pinthudarunnathinaal ithu aavasyamilla. 
nirvachanam anusarichu, formula _ 1-leeii oru maanadandam pooyintukal verthirikkunna oru seminormaanu, athaayath athinu inipparayunna adhika property undu
seminome formula _ 11 umm oru maanadandamaanengil seminome formula _ 13 neey aaaa ennu vilikkunnu.
samboornna ekathaanatha positteev ekathaanathaye soochippikkunnathinaal, oro seminorm oru sublinear function ennariyappedunna oru tharam functionaanu. oru mapp formula _ 4 ennathu subaditive positteev homogeneous aanenkil athine a ennu vilikkunnu. oru seminormil ninnu vyathyasthamayi, oru sublinear function theerchayaayum noonn negatteevu aanu. hahn-banach sidhaanthathinte pashchaathalatthil sublinear functionukal palappozhum abhimukeekarikkappedunnu. 
oru yadhaartha moolyamulla function formula _ 4 oru sublinear balanced function aanenkil maathram oru seminormaanu.
minkovsky functionalukalum seminormukalum.
oru vekttar space formula _ 1 leeii seminormukal minkovsky functionalukal vazhi convex, balanced, absorbing ennee formula _ 1 nte upavibhagangalumaayi adutha bandhikkappettirikkunnu. formula 24-nte atharam oru upavibhaga soothravaakyam nalkiyaal formula 23-nte mingovsky functional oru seminormaanu. neremarichu, formula 24-ll setformula 28 umm formula 29 umm convex, balanced, absorbing ennivayaanu, koodaathe, eee rund settukaludeyum (athupole thanne "avaykkidayil" kidakkunna eathenkilum settinte) mingovsky functional formula 30 aanu.
beejaganita gunangal.
oro seminorm oru sublinear functionaanu, athinaal konveksitti, formula _ 31, ellah vector formula _ 32 ennivayulppede oru sublinear functionte ellah gunangalum thripthippeduthunnu
vipareeta trikona asamathvam 
soothravaakyam _ 33
koodaathe
formula _ 34 umm formula _ 35 umm
eathenkilum vekttar formula _ 8 num positteev riyal formula _ 37 num
soothravaakyam _ 38
koodaathe, formula _ 39 formula _ 16 leeii oru agiranam cheiyunna disk aanu.
formula _ 11 oru riyal vekttar space formula _ 1 leeii oru sublinear functionaanengil, formula _ 1 ll oru leanier functional formula _ 43 nilavilundu, athaayath formula _ 45, koodaathe, formula _ 24 formula _ 48 leeii eathenkilum leanier functional formula _ 46 nuu formula _ 1 ll formula _ 50 aanenkil maathram
seminormukalude mattu savisheshathakal
oro seminorm oru santhulithamaaya pravarthanamaanu. 
formula _ 28ll nissaaramallaattha vekttar subpace adangiyittillengil maathram oru seminome formula _ 11 formula _ 1 leeii oru maanadandamaanu. 
formula _ 54 formula _ 1 leeii oru seminormaanengil formula _ 56 formula _ 1 nte oru vekttar subspace aanu, koodaathe oro formulakkum _ 58 formula _ 11 settu formula _ 60 ll sthiravum formula _ 61 nuu thulyavumaanu. 
koodaathe, eathenkilum yadhaartha soothravaakyathinu _ 62 
soothravaakyam _ 63
formula _ 23 oru koottam thripthikaramaaya formula _ 65 aanenkil, formula _ 23 formula _ 1 lum formula _ 68 lum agiranam cheiyunnu, avide formula _ 69 formula _ 23 (athaayath, formula _ 23-nte gej) mistovskyude pravartthanatthe soochippikkunnu. prathyekichum, formula _ 23 mukalilullathupole aanenkil formula _ 73 formula _ 24 leeii eathenkilum seminormaanengil formula _ 76 aanenkil maathram formula _ 75
formula _ 77 oru normed space aandu formula _ 32 aanenkil, interwell formula _ 81 leeii ellah formulakalkkum formula _ 79 
oro maanadandavum oru convex functionaanu, thalphalamaayi, oru maanadandatthe adisthaanamaakkiyulla vasthunishtamaaya pravartthanatthinte aagola paramavadhi kandethunnathu chilappol tractable aanu.
mattu maanadandangal polulla aashayangalumaayulla bandham.
formula _ 4 oru noonn-negatteevu functionayirikkatte. thaazhepparayunnava thulyamaanu
melpparanja eathenkilum vyavasthakal nilanilkkunnundenkil, inipparayunnava thulyamaanu
formula _ 11 oru riyal vekttar space formula _ 1 leeii oru sublinear functionaanengil inipparayunnava thulyamaanu 
seminormukal ulppedunna asamathvangal.
formula _ 85 formula _ 1 leeii seminormukalaanenkil
formula _ 11 formula _ 1 leeii oru seminorm formula _ 43 formula _ 1 leeii oru leanier functionalumaanenkil
seminormukalkkulla hunn-banach sidhaantham.
seminormukal hahn-banach sidhaanthathinte prathyeka shudhamaaya roopeekaranam vaagdaanam cheiyunnu 
samanamaaya oru vipuleekarana swath seminormukalkkum undu
seminormed speysukalude topologykal.
seudometrics induced topology.
formula _ 1-leeii oru seminome formula _ 11, canonical translation-invariyant seudometric formula _ 113; formula _ 114 vazhi oru topology prerippikkunnu. 
formula _ 115 oru metrik aanenkil maathram eee topology hosdorff aanu, ithu formula _ 11 oru maanadandamaanengil maathrame sambhavikkoo. 
eee topology formula _ 1 neey praadeshikamaayi convex seudometrisable topological vekttar spaceaki maattunnu, athu uthbhavathinte oru parimithamaaya ayalpakkavum uthbhavasthaanath inipparayunna thuranna panthukal (allengil adacha panthukal) ulkkollunna oru ayalpakka aditharayumaanu
soothravaakyam _ 118
formula 119 positteev realukalkku mukalilaanu. 
mattuvidhathil soochippichittillengil oro seminormed space formula _ 13 eee topology nalkiyittundennu karuthanam. chila seminormukalaal prachodippikkappedunna topological vekttar space ennu vilikkunnu. 
thulyamaayi, seminome formula _ 11 ulla oro vekttar space formula _ 1 oru vekttar space kwaasant formula _ 123 indux cheiyunnu, avide formula _ 124 ennathu formula _ 127 ulla ellah vekttar formulakalum _ 8 ulkkollunna formula _ 1 nte subspace aanu, thudarnnu formula _ 128 formula _ 129 nirvachicha oru maanadandam vahikkunnu, thathfalamaayundaakunna topology, formula _ 24 lake valichidunnu, ithu formula _ 30 indues cheytha topologyaanu.
eathenkilum seminome-induced topology formula _ 1 praadeshikamaayi convex akkunnu, inipparayunna reethiyil. formula _ 11 formula _ 1, formula _ 135 ennivayile oru seminormaanu, settu formula _ 39 ennu vilikkuka; athupole radius formula _ 137 nte clossed booal formula _ 138 aanu, ellah ooppan (rep) kaludeyum settu. clossed) formula _ 11-bolukal uthbhavasthaanath thurannirikkunna convex balanced settukalude ayalpakka adisthaanam roopappeduthunnu (rep. clossed) formula _ 11-topology formula _ 16
shakthavum durbalav thulyavumaaya seminormukal.
shakthavum durbalavumaaya seminormukalude ashayangal shakthavum durbalavumaaya maanadandangalude aashayangalkk samaanamaan. formula _ 11 umm formula _ 73 umm formula _ 24 leeii seminormukalaanenkil, formula _ 73 formula _ 11 nekkaal kooduthalaanennum thaazhepparayunna thulyamaaya vyavasthakalil eathenkilum undengil formula _ 11 formula _ 73 nekkaal kooduthalaanennum nammal parayunnu
seminormukalude formula _ 11, formula _ 73 enniva parasparam durbalamaanengil (allengil randum shakthamaanenkil) vilikkunnu. thaazhepparayunna eathenkilum vyavasthakal avar paalikkukayaanenkil ithu sambhavikkunnu
saadhaaranavum semi-ormmabilittiyum.
oru topological vekttar space (tvs) athinte topology orotta seminome (rep) prerithamakkiyaal (yathaakramam, eiii) ennu parayappedunnu. orotta maanadandam). 
oru tii. vii. sii. sii. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. oo. 
uthbhavasthaanathinte oru parimithamaaya ayalpakkamulla oru topological vekttar space aanu a. 
topological vekttar speysukalude maanadandam kolmogorovinte maanadanda maanadandamaanu. 
uthbhavasthaanath oru convex bounded ayalpakkamundengil maathrame oru television seminormable aakoo. 
athinaal oru praadeshikamaayi convex tvkalkk shoonyamallaatha oru parimithikalillaattha ooppan settu undengil maathrame athu ardhasakriyamaakoo.
oru tii. vii. ezz oru oru space aanenkil maathram athu saadhaaranamaanu, koodaathe uthbhavasthaanathinte oru parimithamaaya convex ayalpakkathe amgeekarikkukayum cheiyunnu.
formula _ 1 praadeshikamaayi convex tivikalaanenkil inipparayunnava thulyamaanu 
koodaathe, soothravaakyam 176 saadhaaranamaanenkil maathram formula 1 parimithamaaya alavilaanu (ivide soothravaakyam 176 durbalamaaya topology ulla formula 178 neey soochippikkunnu).
eee edangalil ellaam parimithamaanengil (athaayath, 0-dimensional) maathrame ananthamaaya niravadhi seminormable spacinte uthppannam veendum seminormable aakoo.
seminormukalude thudarcha.
formula _ 11 oru topological vekttar space formula _ 24 leeii oru seminormaanengil inipparayunnava thulyamaanu 
prathyekichum, formula _ 13 oru seminormed space aanenkil, formula _ 73-ll formula _ 30-nte positteev scaler multiple aadhipathyam pulartthukayaanenkil maathrame formula _ 1-leeii oru seminorm formula _ 73 thudarchayaayulloo.
formula _ 1 oru yadhaartha television aanenkil, formula _ 43 formula _ 24-leeii oru leanier functional aanu, formula _ 11 formula _ 24-leeii oru thudarchayaaya seminormaanu (allengil kooduthal saadhaaranayaayi, oru sublinear function), formula _ 1-leeii formula _ 45 formula _ 43 thudarchayaanennu soochippikkunnu.
rekheeya bhoopadangalude thudarcha.
formula _ 194 semi-normed speysukalkkidayilulla oru bhoopadamaanenkil,
formula _ 195
formula _ 194 seminormed speysukalkkidayilulla oru leanier mapp aanenkil inipparayunnava thulyamaanu
formula _ 96 thudarchayaanenkil ellah formulakalkkum formula _ 198
seminormed speysukalkkidayilulla ellah thudarchayaaya leanier mapp formulakalude space _ 194 seminormed formula _ 201 nuu keezhilulla oru seminormed speysaanu. 
formula _ 73 oru maanadandamaanengil eee seminorm oru maanadandamaanu.
saamaanyavalkkaranangal.
ghadanayil algibras enna ashayam oru maanadandatthinte saadhaarana gunangal pankidunnu.
oru composition beejaganita formula _ 203 ll oru feeld formula _ 204 nuu mukalilulla oru beejaganithavum oru involsion formula _ 205 umm quadratic phoom formula _ 206 umm adangiyirikkunnu, athine "norm" ennu vilikkunnu. niravadhi sandarbhangalil formula _ 207 oru isotropic quadratic roopamaanu, athinaal formula _ 208 nuu kuranjathu oru nall vektarenkilum undu, eee lekhanathil charcha cheytha saadhaarana maanadandatthinu aavashyamaaya pooyintukalude verthirikkalin vipareethamaayi.
a allengil a ennathu oru seminome formulayanu _ 4 athu formula thripthippeduthunnu _ 210
durbalamaaya sub aditivitti ardha-seminomukal
oru mapp formula _ 4 (thikachum) ekathaanavum formula _ 212 polulla chila formula nilavilundenkil athine a ennu vilikkunnu.
ithu nilanilkkunna formula _ 214-nte ettavum cheria moolyathe vilikkunnu 
pooyintukal verthirikkunna oru ardha-samaroopathe soothravaakyam _ 16 ll a ennu vilikkunnu.
durbalamaaya ekathaanatha-soothravaakyam _ 216-seminomukal
oru mapp formula _ 4-neey a ennu vilikkunnu, athu subaditive aanenkil, aaa formula _ 219 polulla oru formula _ 216 nilavilundenkil, ellah formulakalkkum _ 8, scalerukal formula _ 221 formula _ 222 ennivaykku pooyintukal verthirikkunna oru formula _ 216-seminormine a oonn formula _ 16 ennu vilikkunnu.
ardha-seminormukalum formula _ 216-seminormukalum thammil namukk inipparayunna bandhamunda
kurippukal.
thelivukal

