For Rp fultkemp/280A/13.2.Lp-Complete-After.pdft.ge LP. Thus LP is a normed vector space. This is...
Transcript of For Rp fultkemp/280A/13.2.Lp-Complete-After.pdft.ge LP. Thus LP is a normed vector space. This is...
Remember LP :
LP or,Tsn) = { feels
,8.y) : fl fl Pdn cos} He - null sets
We've looked carefully at Lt and L2.
More generally :
Theorem : 117.241 For l speos ,
Hf Rp : = ( ful fl Pda )"P
defines a norm onLP Irish) .
In particular :
H ft g Hip S H f the t 11g the
t.ge LP .
Thus LP is a normed vector space .
This is Minkowski's inequality .
We proved it already for p- 12 .
In general , it goes beyond the scope of what we need to prove it .
LP us .
La convergenceLemma : Let fn
, ft LMR,qµ) l l speos ) with Hfn - flip → o .
Then fn→µ f .
Pf . Morbius sina.gl?jt:eI..g*nppqi=Iiifn.fnu → o.
¥: is:": 't.ci.
.
tf lfn He} s an!ffn Ian} =L→ o
.
i. fn→go .
pH
Hfn - other = ftp.n/Pdx--&uPdA--ht-hP--nM.ipfn-olkp-nHpto
Theorem : 117.271 For Isp cos , Lt is Cauchy complete :
fine LP,Hfn - fmllhp To as him → as
⇒ I felt sit . llfn - f the → o .
Pf . H - Cauchy ⇒ Le - Cauchy : →
ME lfn - false}"Em tgpflfn-fmp.de = tlfmfmlkp → e
i.
is E - Cauchy .
E' '
names .
€2 fu .→ f as . , fate .Hfn . - flip Ets - ff lfmifnjlpefr Fasten him!flfna-fnjldnkm.lt) s fnssbggiiptllfn
.
- find. .
LIFT " Kiki the
i. It fn-ftkp-Nfn-fnnltlfna-fdlkpsnfn-fnalkptllfna-flkpg-g.skSo as n, knees
Summary[ convergence
"" iii.✓ convergence
DCTJ
Ip > t) 2- as. convergence
Theorem : ( Lo - L'DCT)
Let fn.gn.ge Lt , fell , s- k
( D lfnlsgn a. s .
(2) f.→af , gn→µg
③ Ign → fgThen ft L' and stfu → e .
( In particular, ffn -off . )Pf. By al , can find fun→ f Agha→g a - s
.
If Isa .by?.lfnalsfF.gne--g a. s .
S -
pose thatf#→ .
Fao, 8 me sit
.ftVk ,
But fun -7 f,i . fna→f as . gna → g a - s .
If - finals Ifl tlfnnlsgtgn.→ 2g . Also flgtgmd '- fgtfgmeosfzg .
i. by DCT ⇒ lff . Nu .i. I Sfnisfl -- Isan -Al sflfnfto .H