:: Continuity of Multilinear Operator on Normed Linear Spaces
:: by Kazuhisa Nakasho and Yasunari Shidama
::
:: Received February 27, 2019
:: Copyright (c) 2019-2021 Association of Mizar Users

theorem NDIFF825: :: LOPBAN11:1
for n being Nat
for r being Real st 0 < r holds
ex s being Real st
( 0 < s & s < r & sqrt ((s * s) * n) < r )
proof end;

theorem LM03: :: LOPBAN11:2
for R1, R2 being FinSequence of REAL
for n, i being Nat
for r being Real st i in dom R1 & R1 = n |-> 1 & R2 = R1 +* (i,r) holds
Product R2 = r
proof end;

theorem :: LOPBAN11:3
for F being FinSequence of REAL st ( for k being Element of NAT st k in dom F holds
0 <= F . k ) holds
0 <= Product F
proof end;

theorem DCARXX: :: LOPBAN11:4
for X being RealNormSpace-Sequence holds dom (carr X) = dom X
proof end;

theorem ZERXI: :: LOPBAN11:5
for X being RealNormSpace-Sequence
for z being Element of () st z = 0. () holds
for i being Element of dom X holds z . i = 0. (X . i)
proof end;

theorem FXZER: :: LOPBAN11:6
for X being RealNormSpace-Sequence
for Y being RealNormSpace
for f being MultilinearOperator of X,Y holds f . (0. ()) = 0. Y
proof end;

theorem PSPROD: :: LOPBAN11:7
for F being FinSequence of REAL st ( for i being Element of dom F holds F . i > 0 ) holds
Product F > 0
proof end;

theorem Th42: :: LOPBAN11:8
for X being RealNormSpace-Sequence
for Y being RealNormSpace st Y is complete holds
for seq being sequence of st seq is Cauchy_sequence_by_Norm holds
seq is convergent
proof end;

theorem Th43: :: LOPBAN11:9
for X being RealNormSpace-Sequence
for Y being RealBanachSpace holds R_NormSpace_of_BoundedMultilinearOperators (X,Y) is RealBanachSpace
proof end;

registration
let X be RealNormSpace-Sequence;
let Y be RealBanachSpace;
coherence by Th43;
end;

theorem NDIFF823: :: LOPBAN11:10
for n being Nat
for F being Element of REAL n
for s being Real st ( for i being Nat st i in dom F holds
( 0 <= F . i & F . i <= s ) ) holds
|.F.| <= sqrt ((s * s) * (len F))
proof end;

theorem LM02: :: LOPBAN11:11
for X being RealNormSpace-Sequence
for Y being RealNormSpace
for f being MultilinearOperator of X,Y
for K being Real st 0 <= K & ( for x being Point of () holds ||.(f . x).|| <= K * () ) holds
for v0, v1 being Point of ()
for Cv0, Cv1 being FinSequence
for i being Element of dom X st Cv0 = v0 & Cv1 = v1 & ||.(v1 - v0).|| <= 1 & ( for j being Element of dom X st i <> j holds
Cv1 . j = Cv0 . j ) holds
||.((f /. v1) - (f /. v0)).|| <= (((||.v0.|| + 1) |^ (len X)) * K) * ||.((v1 - v0) . i).||
proof end;

theorem LM01: :: LOPBAN11:12
for X being RealNormSpace-Sequence
for Y being RealNormSpace
for f being MultilinearOperator of X,Y
for K being Real st 0 <= K & ( for x being Point of () holds ||.(f . x).|| <= K * () ) holds
for v0 being Point of () ex M being Real st
( 0 <= M & ( for v1 being Point of () st ||.(v1 - v0).|| <= 1 holds
ex F being FinSequence of REAL st
( dom F = dom X & ||.((f /. v1) - (f /. v0)).|| <= (M * K) * (Sum F) & ( for i being Element of dom X holds F . i = ||.((v1 - v0) . i).|| ) ) ) )
proof end;

theorem NDIFF824: :: LOPBAN11:13
for X being RealNormSpace-Sequence
for x being Point of ()
for r being Real st 0 < r holds
ex s being FinSequence of REAL ex Y being non-empty non empty FinSequence st
( dom s = dom X & dom Y = dom X & product Y c= Ball (x,r) & ( for i being Element of dom X holds
( 0 < s . i & s . i < r & Y . i = Ball ((x . i),(s . i)) ) ) )
proof end;

theorem :: LOPBAN11:14
for X being RealNormSpace-Sequence
for Y being RealNormSpace
for f being MultilinearOperator of X,Y holds
( ( f is_continuous_on the carrier of () implies f is_continuous_in 0. () ) & ( f is_continuous_in 0. () implies f is_continuous_on the carrier of () ) & ( f is_continuous_on the carrier of () implies f is Lipschitzian ) & ( f is Lipschitzian implies f is_continuous_on the carrier of () ) )
proof end;