:: Weak Convergence and Weak* Convergence
:: by Keiko Narita , Noboru Endou and Yasunari Shidama
::
:: Received July 1, 2015
:: Copyright (c) 2015-2019 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies RLSUB_1, HAHNBAN, UNIALG_1, DUALSP01, DUALSP02, DUALSP03, PBOOLE,
INT_1, NFCONT_1, CFCONT_1, RLVECT_1, ARYTM_1, RELAT_1, FUNCT_1, RSSPACE,
RSSPACE3, REAL_1, TARSKI, MSSUBFAM, STRUCT_0, SEQ_1, FUNCOP_1, FCONT_1,
SEQ_2, LOPBAN_1, ORDINAL2, ARYTM_3, NORMSP_1, FUNCT_2, PRE_TOPC,
SUBSET_1, ZFMISC_1, NUMBERS, SEQ_4, XBOOLE_0, CARD_1, SUPINF_2, COMPLEX1,
XXREAL_0, XXREAL_2, NAT_1, REWRITE1, VALUED_1, METRIC_1, ASYMPT_1,
PARTFUN1, RCOMP_1, NORMSP_2, RLVECT_3, NORMSP_3, TOPS_1, TOPGEN_1,
VALUED_0;
notations TARSKI, XBOOLE_0, SUBSET_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1,
FUNCT_2, FUNCOP_1, NUMBERS, XCMPLX_0, XXREAL_0, XREAL_0, NAT_1, INT_1,
VALUED_0, COMPLEX1, XXREAL_2, VALUED_1, SEQ_1, SEQ_2, SEQ_4, RINFSUP1,
STRUCT_0, PRE_TOPC, TOPS_1, RLVECT_1, RLSUB_1, RLVECT_3, NORMSP_0,
NORMSP_1, HAHNBAN, RSSPACE, RSSPACE3, LOPBAN_1, NFCONT_1, NORMSP_2,
LOPBAN_5, DUALSP01, NORMSP_3, DUALSP02;
constructors REAL_1, EUCLID, REALSET1, RSSPACE3, BINOP_2, SEQ_1, NFCONT_1,
RELSET_1, SEQ_2, SEQ_4, HAHNBAN1, NORMSP_2, RLVECT_3, LOPBAN_5, COMSEQ_2,
NORMSP_3, TOPS_1, DUALSP02, RINFSUP1, CARD_3;
registrations STRUCT_0, XREAL_0, NUMBERS, ORDINAL1, MEMBERED, RELAT_1,
XXREAL_0, VALUED_0, FUNCT_2, VALUED_1, FUNCOP_1, SEQ_4, RELSET_1,
FUNCT_1, NORMSP_3, DUALSP02, XCMPLX_0, NORMSP_0, NAT_1, NORMSP_1, SEQ_2,
DUALSP01, XBOOLE_0, SEQM_3, INT_1, LOPBAN_1, PRE_TOPC, NORMSP_4;
requirements NUMERALS, BOOLE, SUBSET, REAL, ARITHM;
begin :: Some properties about Dual Spaces of Real Normed Spaces
definition
let X be non empty set;
let F be sequence of Funcs(NAT,X);
let k be Nat;
redefine func F.k -> sequence of X;
end;
theorem :: DUALSP03:1
for X be strict RealNormSpace, A be non empty Subset of X holds
(for f be Point of DualSp X st
(for x be Point of X st x in A holds (Bound2Lipschitz(f,X)).x = 0)
holds Bound2Lipschitz(f,X) = 0.(DualSp X))
implies ClNLin(A) = X;
theorem :: DUALSP03:2
for X be strict RealNormSpace st DualSp X is separable holds X is separable;
theorem :: DUALSP03:3
for x be Real, x1 be Point of RNS_Real st x=x1 holds -x = -x1;
theorem :: DUALSP03:4
for x,y be Real, x1,y1 be Point of RNS_Real
st x=x1 & y=y1 holds x-y = x1-y1;
theorem :: DUALSP03:5
for seq be Real_Sequence, seq1 be sequence of RNS_Real
st seq = seq1 holds seq is convergent iff seq1 is convergent;
theorem :: DUALSP03:6
for seq be Real_Sequence, seq1 be sequence of RNS_Real
st seq = seq1 & seq is convergent holds lim seq = lim seq1;
theorem :: DUALSP03:7
for seq1 be sequence of RNS_Real
st seq1 is Cauchy_sequence_by_Norm holds seq1 is convergent;
registration
cluster RNS_Real -> complete;
end;
definition
let X be RealNormSpace,
g be sequence of DualSp X,
x be Point of X;
func g#x -> Real_Sequence means
:: DUALSP03:def 1
for i be Nat holds it.i = (g.i).x;
end;
begin :: Weak Convergence and Weak* Convergence
definition
let X be RealNormSpace, x be sequence of X;
attr x is weakly-convergent means
:: DUALSP03:def 2
ex x0 be Point of X st
for f be Lipschitzian linear-Functional of X
holds f*x is convergent & lim (f*x) = f.x0;
end;
theorem :: DUALSP03:8
for X be RealNormSpace, x be sequence of X
st rng x c= {0.X} holds x is weakly-convergent;
definition
let X be RealNormSpace, x be sequence of X;
assume x is weakly-convergent;
func w-lim x -> Point of X means
:: DUALSP03:def 3
for f be Lipschitzian linear-Functional of X
holds f*x is convergent & lim (f*x) = f.it;
end;
theorem :: DUALSP03:9
for X be RealNormSpace, x be sequence of X
st x is convergent holds x is weakly-convergent & w-lim x = lim x;
theorem :: DUALSP03:10
for X be RealNormSpace, x be sequence of X
st X is non trivial & x is weakly-convergent
holds ||.x.|| is bounded & ||. w-lim x .|| <= lim_inf ||.x.||
& w-lim x in ClNLin(rng x);
definition
let X be RealNormSpace,
g be sequence of DualSp X;
attr g is weakly*-convergent means
:: DUALSP03:def 4
ex g0 be Point of DualSp X st
for x be Point of X holds g#x is convergent & lim (g#x) = g0.x;
end;
definition
let X be RealNormSpace,
g be sequence of DualSp X;
assume g is weakly*-convergent;
func w*-lim g -> Point of DualSp X means
:: DUALSP03:def 5
for x be Point of X
holds g#x is convergent & lim (g#x) = it.x;
end;
theorem :: DUALSP03:11
for X be RealNormSpace, g be sequence of DualSp X
st g is convergent holds g is weakly*-convergent & w*-lim g = lim g;
theorem :: DUALSP03:12
for X be RealNormSpace, f be sequence of DualSp X
st f is weakly-convergent holds f is weakly*-convergent;
theorem :: DUALSP03:13
for X be RealNormSpace, f be sequence of DualSp X
st X is Reflexive holds
f is weakly-convergent iff f is weakly*-convergent;
theorem :: DUALSP03:14
for X be RealBanachSpace, T be Subset of DualSp X
st ( for x be Point of X
ex K be Real st
0 <= K
& for f be Point of DualSp X st f in T holds |. f.x .| <= K ) holds
ex L be Real st
0 <= L
& for f be Point of DualSp X st f in T holds ||.f.|| <= L;
theorem :: DUALSP03:15
for X be RealBanachSpace, f be sequence of DualSp X
st f is weakly*-convergent
holds ||.f.|| is bounded & ||. w*-lim f .|| <= lim_inf ||.f.||;
theorem :: DUALSP03:16
for X be RealNormSpace, x be Point of X,
vseq be sequence of DualSp X,
vseq1 be sequence of R_NormSpace_of_BoundedLinearOperators(X,RNS_Real)
st vseq = vseq1 holds vseq#x = vseq1#x;
theorem :: DUALSP03:17
for X be RealBanachSpace, X0 be Subset of X, vseq be sequence of DualSp X
st ||.vseq.|| is bounded & X0 is dense
& (for x be Point of X st x in X0 holds vseq#x is convergent)
holds vseq is weakly*-convergent;
theorem :: DUALSP03:18
for X be RealBanachSpace, f be sequence of DualSp X holds
f is weakly*-convergent
iff ( ||.f.|| is bounded
& ex X0 be Subset of X st
X0 is dense
& (for x be Point of X st x in X0 holds f#x is convergent)
);
begin :: Weak Sequential Compactness of Real Banach Spaces
definition
let X be RealNormSpace, X0 be non empty Subset of X;
attr X0 is weakly-sequentially-compact means
:: DUALSP03:def 6
for seq be sequence of X0
ex seq1 be sequence of X st
seq1 is subsequence of seq & seq1 is weakly-convergent
& w-lim seq1 in X;
end;
theorem :: DUALSP03:19
for X be RealNormSpace, x be sequence of X st X is Reflexive holds
x is weakly-convergent iff (BidualFunc X)*x is weakly*-convergent;
theorem :: DUALSP03:20
for X be RealNormSpace, f be sequence of DualSp X, x be Point of X
st ||.f.|| is bounded holds
ex f0 be sequence of DualSp X st
f0 is subsequence of f & ||.f0.|| is bounded & f0#x is convergent;
theorem :: DUALSP03:21
for X be RealNormSpace, f be sequence of DualSp X, x be Point of X
st ||.f.|| is bounded holds
ex f0 be sequence of DualSp X st
f0 is subsequence of f & ||.f0.|| is bounded
& f0#x is convergent & f0#x is subsequence of f#x;
theorem :: DUALSP03:22
for X be RealNormSpace, f be sequence of DualSp X, x be Point of X
st ||.f.|| is bounded holds
ex f0 be sequence of DualSp X, N be increasing sequence of NAT st
f0 is subsequence of f & ||.f0.|| is bounded
& f0#x is convergent & f0#x is subsequence of f#x
& f0 = f*N;
theorem :: DUALSP03:23
for X be RealNormSpace, f be sequence of DualSp X, x be sequence of X
st ||.f.|| is bounded holds
ex F be sequence of Funcs(NAT,the carrier of DualSp X) st
F.0 is subsequence of f & (F.0)#(x.0) is convergent
& ( for k be Nat holds F.(k+1) is subsequence of F.k )
& ( for k be Nat holds (F.(k+1))#(x.(k+1)) is convergent);
theorem :: DUALSP03:24
for X be RealNormSpace, f be sequence of DualSp X, x be sequence of X
st ||.f.|| is bounded holds
ex F be sequence of Funcs(NAT,the carrier of DualSp X),
N be sequence of Funcs(NAT,NAT) st
F.0 is subsequence of f & (F.0)#(x.0) is convergent
& N.0 is increasing sequence of NAT & F.0 = f*N.0
& (for k be Nat holds F.(k+1) is subsequence of F.k)
& (for k be Nat holds (F.(k+1))#(x.(k+1)) is convergent)
& (for k be Nat holds (F.(k+1))#(x.(k+1)) is subsequence of (F.k)#(x.(k+1)))
& (for k be Nat holds N.(k+1) is increasing sequence of NAT)
& for k be Nat holds F.(k+1) = (F.k)*N.(k+1);
theorem :: DUALSP03:25
for X be RealNormSpace, f be sequence of DualSp X, x be sequence of X
st ||.f.|| is bounded holds
ex M be sequence of DualSp X st
M is subsequence of f & for k be Nat holds M#(x.k) is convergent;
theorem :: DUALSP03:26
for X be RealBanachSpace, f be sequence of DualSp X
st X is separable & ||.f.|| is bounded holds
ex f0 be sequence of DualSp X st
f0 is subsequence of f & f0 is weakly*-convergent;
theorem :: DUALSP03:27
for X be RealBanachSpace, x be sequence of X
st X is Reflexive & ||.x.|| is bounded holds
ex x0 be sequence of X st
x0 is subsequence of x & x0 is weakly-convergent;
theorem :: DUALSP03:28
for X be RealBanachSpace, X0 be non empty Subset of X
st X is non trivial Reflexive holds
X0 is weakly-sequentially-compact iff
ex S be non empty Subset of REAL st
S = {||.x.|| where x is Point of X : x in X0}
& S is bounded_above;