begin
:: deftheorem Def1 defines are_commutative LOPBAN_4:def 1 :
for X being non empty multMagma
for x, y being Element of X holds
( x,y are_commutative iff x * y = y * x );
Lm1:
for X being Banach_Algebra
for z being Element of X
for n being Element of NAT holds
( z * (z #N n) = z #N (n + 1) & (z #N n) * z = z #N (n + 1) & z * (z #N n) = (z #N n) * z )
Lm2:
for X being Banach_Algebra
for n being Element of NAT
for z, w being Element of X st z,w are_commutative holds
( w * (z #N n) = (z #N n) * w & z * (w #N n) = (w #N n) * z )
theorem Th1:
theorem Th2:
theorem Th3:
theorem Th4:
theorem Th5:
theorem Th6:
theorem Th7:
theorem Th8:
theorem Th9:
theorem Th10:
theorem Th11:
theorem Th12:
:: deftheorem Def2 defines rExpSeq LOPBAN_4:def 2 :
for X being Banach_Algebra
for z being Element of X
for b3 being sequence of X holds
( b3 = z rExpSeq iff for n being Element of NAT holds b3 . n = (1 / (n !)) * (z #N n) );
theorem Th13:
:: deftheorem Def3 defines Coef LOPBAN_4:def 3 :
for n being Element of NAT
for b2 being Real_Sequence holds
( b2 = Coef n iff for k being Element of NAT holds
( ( k <= n implies b2 . k = (n !) / ((k !) * ((n -' k) !)) ) & ( k > n implies b2 . k = 0 ) ) );
:: deftheorem Def4 defines Coef_e LOPBAN_4:def 4 :
for n being Element of NAT
for b2 being Real_Sequence holds
( b2 = Coef_e n iff for k being Element of NAT holds
( ( k <= n implies b2 . k = 1 / ((k !) * ((n -' k) !)) ) & ( k > n implies b2 . k = 0 ) ) );
:: deftheorem Def5 defines Shift LOPBAN_4:def 5 :
for X being non empty ZeroStr
for seq, b3 being sequence of X holds
( b3 = Shift seq iff ( b3 . 0 = 0. X & ( for k being Element of NAT holds b3 . (k + 1) = seq . k ) ) );
:: deftheorem Def6 defines Expan LOPBAN_4:def 6 :
for n being Element of NAT
for X being Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Expan (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = (((Coef n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );
:: deftheorem Def7 defines Expan_e LOPBAN_4:def 7 :
for n being Element of NAT
for X being Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Expan_e (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = (((Coef_e n) . k) * (z #N k)) * (w #N (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );
:: deftheorem Def8 defines Alfa LOPBAN_4:def 8 :
for n being Element of NAT
for X being Banach_Algebra
for z, w being Element of X
for b5 being sequence of X holds
( b5 = Alfa (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = ((z rExpSeq) . k) * ((Partial_Sums (w rExpSeq)) . (n -' k)) ) & ( n < k implies b5 . k = 0. X ) ) );
:: deftheorem Def9 defines Conj LOPBAN_4:def 9 :
for X being Banach_Algebra
for z, w being Element of X
for n being Element of NAT
for b5 being sequence of X holds
( b5 = Conj (n,z,w) iff for k being Element of NAT holds
( ( k <= n implies b5 . k = ((z rExpSeq) . k) * (((Partial_Sums (w rExpSeq)) . n) - ((Partial_Sums (w rExpSeq)) . (n -' k))) ) & ( n < k implies b5 . k = 0. X ) ) );
theorem Th14:
theorem Th15:
theorem Th16:
theorem Th17:
theorem Th18:
theorem Th19:
theorem Th20:
theorem Th21:
theorem Th22:
theorem Th23:
theorem Th24:
theorem Th25:
theorem Th26:
theorem Th27:
theorem Th28:
theorem Th29:
theorem Th30:
theorem Th31:
theorem Th32:
theorem Th33:
Lm3:
for X being Banach_Algebra
for z, w being Element of X st z,w are_commutative holds
(Sum (z rExpSeq)) * (Sum (w rExpSeq)) = Sum ((z + w) rExpSeq)
:: deftheorem Def10 defines exp_ LOPBAN_4:def 10 :
for X being Banach_Algebra
for b2 being Function of the carrier of X, the carrier of X holds
( b2 = exp_ X iff for z being Element of X holds b2 . z = Sum (z rExpSeq) );
:: deftheorem defines exp LOPBAN_4:def 11 :
for X being Banach_Algebra
for z being Element of X holds exp z = (exp_ X) . z;
theorem
theorem Th35:
theorem
theorem Th37:
theorem Th38:
theorem
theorem Th40:
theorem