
sthaaniya roop seey ekikrit kaary

ganit mein, ack sthaaniya roop seey samakalan yogya falan (jise kabhi-kabhi sthaaniya roop seey yog yogya falan bhee kahaa jaataa haye) ack aisaa falan haye joe apane paribhasha key kshetra key pratyek sankshipt upasamuchchay parr samakalan yogya (isaliye iska samakalan seemit haye) haye. iss tarah key karyon kaa mahatva iss tathya mein nihit haye kii unakaa kaary sthaan rikt sthaan key samaan haye, lekin isake sadasyon koo apane kshetra kii seema parr apane vyavhaar parr kissy bhee vikas pratibandh koo santusht karane kii aavashyakta naheen haye (anant parr yadi kshetra aseemit haye): dusare shabdon mein, sthaaniya roop seey ekikrit kaary kshetra kii seema parr manmaane dhang seey teji seey badh sakate hain, lekin phir bhee samanya ekikrit karyon key samaan tareeke seey prabandhit kiye jaa sakate hain.
paribhasha.
maanak paribhasha.
. euclidean space formula _ 1 mein ack khula sett honey dein aur ack lebzue maapane yogya kaary honey dein. aggar aisaa haye
aaee.ee. iska labsgue integral sabhi compact subset parr seemit haye, phir ise "sthaaniya roop seey integrable" kahaa jaataa haye. aise sabhi karyon key samooh koo iss prakaar darshaaya jaataa haih
jahan sootra _ 4 sett key pratibandh koo darshaata haye.
sthaaniya roop seey samakalan yogya falan kii shaastriya paribhasha mein kewal saiddhaantik aur sthalvaigyaanik avadhaaranaon koo maapana shamil haye aur ack sthalvaigyaanik maap sthaan parr jatil-moolya vaale karyon koo amoort roop seey leey jaaya jaa sakata haih haalaanki, choonki iss tarah key karyon kaa sabse aam anuprayog euclidean rikt sthaan parr vitaran siddhant key liye haye, isaliye isamen sabhi paribhaashaaen aur nimnalikhit khand spasht roop seey kewal iss mahatvapurna mamle seey nipatate hain.
ack vaikalpik paribhasha.
. euclidean space formula _ 1 mein ack khula sett honey dein. phir ack aisaa kaary joe
pratyek pareekshan falan key liye "sthaaniya roop seey samakalan yogya" kahaa jaataa haye, aur aise karyon key samooh koo isake dwara darshaaya jaataa haye. yahaan sabhi aseem roop seey avakalaniya karyon key samooh koo sandarbhit kartaa haye jisamen compact samarthan nihit haye.
iss paribhasha kii jadein ack topological vector space parr nirantar raikhik karyatmak kii avadharana parr adharit maap aur ekikaran siddhant key drishtikon mein hain, jise nicolaus borbaki school dwara vikasit kiya gayaa haih yeh bhee ack haye jise aur dwara apanaaya gayaa haye. yeh "vitaran saiddhaantik" paribhasha maanak key barabar haye, jaisa kii nimnalikhit shabdansh saabit kartaa haih
. ack diyaa gayaa falan sthaaniya roop seey ekikrit haye yadi aur kewal tabhi jaba yeh sthaaniya roop seey ekikrit haye, i key anusaar.ee.
pramaan.
yadi bhaagah ack pareekshan kaary honey dein. yeh apane sarvocch maanak seey ghiraa huaa haye, maapane yogya haye, aur ack compact samarthan haye, ise coll karein. isaliye
dwara.
kewal aggar hissaah khula sett kaa ack compact subset honey dein. humm pahale ack pareekshan kaary kaa nirmaan karenge joe isake sanketak kaary koo pramukh banaata haye.
seema aur seema key beach samanya nirdhaarit doori shoonya seey adhik haye, i.ee.
isaliye ack vaastavik sankhya kaa chayan karna sambhav haye (yadi khaali sett haye, tou lein). kramshah bund-padosi aur-padosi koo darshaate hain. way bhee isi tarah seey saghan aur santusht hain.
abb falan koo paribhaashit karane key liye convolution kaa upayog karein
jahan maanak sakaratmak samamit kaa upayog karke ack molifier kaa nirmaan kiya jaataa haye. spasht roop seey iss arth mein gair-nakaratmak haye kii, aseem roop seey vibhedak, aur iska samarthan nihit haye, vishesh roop seey yeh ack pareekshan kaary haye. kyonki sabhi key liye, hamaare paas vaha haye.
maan leejiye kii yeh sthaaniya roop seey ekikrit kaary haye. taba
choonki yeh pratyek compact subset key liye dhaaran kartaa haye, isaliye kaary sthaaniya roop seey ekikrit haye. .
saamaanyikaranah sthaaniya roop seey "pee"-ekikrit kaary.
. euclidean space formula _ 1 mein ack khula sett honey dein aur formula _ 14 ack lebzue maapane yogya kaary hoe. yadi, diye gaye key liye, santusht kartaa haye
aaee.ee.yeh sabhi compact upasamuchchay key liye haye, phir ise "sthaaniya roop seey"-"samakalan yogya" yaa "sthaaniya roop seey samakalan yogya" bhee kahaa jaataa haye. aise sabhi karyon key samooh koo iss prakaar darshaaya jaataa haih
ack vaikalpik paribhasha, joe sthaaniya roop seey samakalan yogya karyon key liye dee gaee paribhasha key poori tarah seey anuroop haye, sthaaniya roop seey samakalan yogya karyon key liye bhee dee jaa sakati haih yeh iss khand mein ack key barabar bhee hoe sakati haye aur saabit hoe sakati haye. unaki spasht uchch vyaapaktaa key bavjood, sthaaniya roop seey ekikrit kaary pratyek aise kaary key liye sthaaniya roop seey ekikrit karyon kaa ack upasamuchchay banaate hain.
sanketan.
vibhinn gliff key alawa, jinka upayog badey akshar "ell" key liye kiya jaa sakata haye, sthaaniya roop seey samakalan karyon key samooh key sanketan key liye kuchh prakaar hain.
guna.
"l" "p", "lock sabhi" "p" ≤1 key liye ack poorna metrick sthaan haye. 
. ack poorna metrizable sthaan haih isaki topology nimnalikhit metrick dwara utpann kii jaa sakati haih
jahan khaali naheen honey vaale aise seton kaa parivaar haye
sandarbhon mein, aur, yeh pramey kahaa gayaa haye lekin aupchaarik aadhar parr saabit naheen huaa haih ack adhik samanya parinam kaa ack poorna pramaan, jisamen yeh shamil haye, paaya jaataa haye.
"l" "p" "" "l" "1 kaa ack upasthaan haye, joe sabhi" "p" "≤1 key liye sthaan haye.
. jahan sootra _ 1 kaa ack khula upasamuchchay haye, vahaan seey sambandhit pratyek falan sthaaniya roop seey samakalan yogya haye.
pramaan. mamla tuchchh haye, isaliye saboot kii agali kadee mein yeh maanaa jaataa haye. ack sankshipt upasamuchchay key vishisht kaary parr vichaar keejiyeh tou, key liye,
kahaan
phir kissy bhee sambandhit key liye, holder kii asamaanta key anusaar, utpaad ekikrit haye.ee. aur seey sambandhit haye
isaliye
dhyaan dein kii choonki nimnalikhit asamaanta sach haye
yeh pramey kewal sthaaniya roop seey ekikrit karyon key sthaan seey sambandhit karyon key liye bhee sahee haye, isaliye pramey nimnalikhit parinam kaa bhee tatparya haye.
. pratyek falan sootra _ 29 sootra _ 30, sootra _ 31 mein.
notah yadi sootra _ 32 sootra _ 33 kaa ack khula upasamuchchay haye joe bhee baadhya haye, tou ack key paas maanak samaveshan sootra _ 34 haye joe uparokt samavesh sootra _ 35 koo dekhate hue samajh mein aataa haye. lekin inn bayanon mein seey pahala sahee naheen haye yadi sootra _ 32 baadhya naheen haye; tou yeh abhee bhee sach haye kii sootra _ 37 kissy bhee sootra _ 38 key liye, lekin usa sootra _ 39 key liye naheen haye. ise dekhane key liye, koee bhee aamtaur parr falan sootra _ 40 parr vichaar kartaa haye, joe sootra _ 41 mein haye, lekin kissy bhee seemit sootra _ 38 key liye sootra _ 42 mein naheen haye.
"l" 1, lock bilkul nirantar upaayon key ghanatva kaa sthaan haye.
. ack falan ack purnataya nirantar maap kaa ghanatva haye yadi aur kewal yadi sootra _ 44 haye.
iss parinam kaa pramaan rekhaachitrit kiya gayaa haye. apane kathan koo dohrate hue, yeh pramey iss baat parr jore detaa haye kii pratyek sthaaniya roop seey samakalan yogya falan ack poori tarah seey nirantar maap koo paribhaashit kartaa haye aur isake vipareet pratyek poori tarah seey nirantar upaay ack sthaaniya roop seey samakalan yogya falan koo paribhaashit kartaa haih yeh amoort maap siddhant dhaanche mein bhee haye, joe stanislav sakk dwara apane granth mein diye gaye mahatvapurna raydon-nikodam pramey kaa roop haye.
anuprayog.
sthaaniya roop seey ekikrit kaary vitaran siddhant mein ack pramukh bhoomika nibhaate hain aur way vibhinn vargon key karyon aur kaary sthaanon kii paribhasha mein hotey hain, jaise kii baadhya bhinnata key kaary. isake alawa, way harr maap key bilkul nirantar bhaag koo chihnit karke raydon-nikodam pramey mein dikhaai dete hain.

