metrik space

ganitamlo, metrik space anedi daani muulakaala madhya "dhooram" anae bhaavanatho kuudina samiti, dheenini saadhaaranamgaa paayintlu anie pilustaaru. dhooram anedi metrik ledhaa distains function anie piluvabade function dwara kolustaaru. ganita vislaeshanha mariyu jyaamiti yokka aneka bhaavanalanu adhyayanam cheyadaaniki metrik khaaliilu atyanta saadhaarana amarika.
metrik space yokka atyanta suparichitamaina udaaharana 3-dimensional euclidean space, daani saadhaarana dhooram yokka bhaavanatho. itara prasiddha udaaharanalu koneeya dhooram mariyu hyperbolic plain kaligi unna golam. oka metrik bhouthikamaina, dhooram yokka bhavanaku badulugaa oka roopakaaniki anugunamgaa undavachuhu udaaharanaku, 100-aksharaala unicode teegala samitini hamming dhooramtho amarchavacchu, idhi oka theega nundi maroka teegaku cherukovadaniki maarchaalsina aksharaala sankhyanu kolustundi.
avi chaalaa saadhaaranamainavi kabatti, metrik khaaliilu ganita saastramlooni aneka vibhina saakhalalo upayoginche sadhanam. aneka rakaala ganita vastuvulu dhooram yokka sahaja bhavananu kaligi untaayi, anduvalla remanian manifolds, praamaanika vector spacelu mariyu graphlatho sahaa metrik space yokka nirmaanaanni angeekaristaayi. nairoopya beejaganitamlo, "p"-addic sankhyalu hetubaddha sankhyalapai metrik nirmaanam poortayye mulakaalugaa talettutaayi. metrik sthalaalu kooda metrik jyaamitiloo mariyu metrik pradaesaalapai vislaeshanhaloo vaati swantha hakkulo adhyayanam cheyabadataayi.
ganita vislaeshanha yokka aneka praathamika bhaavanalanu, bantulu, paripuurnata, alaage ekareeti, pedavula pattudala mariyu peddadi konasagimpu vanti vaatini metrik pradaesaala amarikalo nirvachinchavacchu. konasagimpu, compactness, open mariyu closed setlu vanti itara bhaavanalanu metrik spacela kosam nirvachinchavacchu, conny topological spacela marinta saadhaarana amarikalo kooda nirvachinchavacchu.
nirvachanam mariyu drushtaantam.
prerana.
dhooram yokka vividha bhavanala prayojanaanni chuudataaniki, bhoomi yokka uparitalaanni paayintla samitigaa pariganinchandi. kaagam egurutunnanduna, uparitalam venta unna atichinna maargam yokka podavu dwara atuvanti rendu paayintla madhya dooraanni manam kolavavacchu; idhi ravana mariyu vimaanayaanaaniki pratyaekamgaa upayogapadutundi. bhoomi lopali bhaagamlo rendu paayintla madhya sarala rekha dooraanni kooda manam kolavavacchu; udaaharanaku, bhookampa saastramlo eee bhavana sahajamainadi, endukante idhi bhookampa taramgaalu aaa rendu paayintla madhya prayaaninchadaaniki patte samayaniki sumaruga anugunamgaa untundi.
metrik space suutraala dwara encode cheyabadina dhooram anae bhavana saapekshamgaa takkuva avasaralanu kaligi untundi. eee saadhaaraneekarana metrik pradaesaalaku chaalaa vasyatanu istundi. adhe samayamlo, dhooram antey emiti anae daani gurinchi aneka sahajamaina vaastavaalanu encode cheyadaaniki eee bhavana balamgaa undhi. dheeni ardham metrik spacela gurinchi saadhaarana phalitaalanu aneka vibhina sandarbhaallo vartimpajeyavachu.
aneka praathamika ganita bhavanala maadirigaane, metrik sthalamlo metriknu aneka vidhaluga ardham chesukovacchu. oka nirdishta metrik bhouthika dooraanni kolichedigaa uttamamgaa bhavinchakapovacchu, conny, badulugaa, oka rashtram nundi maroka raashtraaniki maaradaaniki ayye kharchu (kolatala pradaesaalapai wasserstein kolamanala maadirigaa) ledhaa rendu vastuvula madhya vyatyaasam yokka degree (udaaharanaku, aksharaala rendu teegala madhya hamming dhooram, ledhaa metrik pradaesaala madhya gromov-husdorf dhooram).
nirvachanam.
adhikarikamga, metrik space anedi oka kramabaddheekarinchina jatha, ikkada oka sett mariyu oka metrik, i.ii.oka function
suutram _ 1
anni paayintla suutram _ 2 kosam eee krindi suutraalanu santruptiparachadamha
metrik nissandehamgaa unte, okati tarachugaa sangnaamaanaanni durviniyogam cheyadam dwara "metrik space" gaa suuchistumdi.
saralamaina udaaharanalu.
nijamaina sankhyalu.
khachitamaina vyatyaasam dwara icchina dhooram function suutram _ 7 thoo vaastava sankhyalu metrik spacenu yerparustaayi. metrik khaaliilu mariyu vaati madhya vidhula yokka aneka lakshanaalu nijamaina vislaeshanhaloo bhavanala saadhaaraneekaranalu mariyu nijamaina rekhaku vartimpajesinappudu aaa bhaavanalatho samanamga untaayi.
euclidean pradaesaalapai kolamanalu.
euclidean plain formula _ 8 aneka vibhina kolamaanaalato amarchabadi untundi. paatasaala ganita shaastram nundi telisina euclidean dooraanni dheeni dwara nirvachinchavacchu
suutram _ 9
taxicab" ledhaa "manhatton" dhooram nirvachinchabadindi
suutram _ 10
mariyu vyaasam eguvana vivarinchinatlugaa, oka point nundi maroka paayintku cherukovadaniki meeru kshitija samaantara mariyu niluvu rekhala venta prayaaninchaalsina dhooram anie bhaavinchavachchu.
garishta", suutram _ 11, ledhaa "chebishev dhooram" nirvachinchabadindi
suutram _ 12) endukante adhi suutram _ 13 anedi banach space anae vaastavam ayithe adhi kalustundani (oka vairudhyam) suuchistumdi.
kolamanala samaanatvam.
icchina sett "x" kosam, rendu kolamaana suutraalu _ 14 mariyu suutram _ 15 nuu gurtimpu mapping ayithe "sthalasaastraparamgaa samanam" ("ekareeti samanam") anie pilustaaru.
idhi oka homomorphism (ekareeti isomorphism).
udaaharanaku, suutram _ 17 oka metrik ayithe, suutram _ 18 mariyu suutram _ 19 suutram _ _ samaanamaina kolamanalu.

