Quelle cal_L_inf.pvs Sprache: PVS

%------------------------------------------------------------------------------
% L^\infty
%
%     Author: David Lester, Manchester University
%
% All references are to SK Berberian "Fundamentals of Real Analysis",
% Springer, 1991
%
% Definition and basic properties of measurable functions f: [T->complex]
%
%     Version 1.0            11/3/10   Initial Version
%------------------------------------------------------------------------------

cal_L_inf[(IMPORTING measure_integration@subset_algebra_def,
                     measure_integration@measure_def)
          T:TYPE, S:sigma_algebra[T], mu:measure_type[T,S]]: THEORY

BEGIN

  IMPORTING essentially_bounded[T,S,mu],
            complex_alt@complex_fun_ops[T]

  h: VAR [T->complex]
  x: VAR T

  cal_L_infty?(h):bool = essentially_bounded?(h)

  cal_L_infty: TYPE+ = (cal_L_infty?) CONTAINING (lambda x: complex_(0,0))

  cal_L_infty_is_essentially_bounded:
                           JUDGEMENT cal_L_infty SUBTYPE_OF essentially_bounded

  f,f0,f1: VAR cal_L_infty
  c: VAR complex

  scal_cal_L_infty: JUDGEMENT *(c,f)   HAS_TYPE cal_L_infty
  add_cal_L_infty:  JUDGEMENT +(f0,f1) HAS_TYPE cal_L_infty
  opp_cal_L_infty:  JUDGEMENT -(f)     HAS_TYPE cal_L_infty
  diff_cal_L_infty: JUDGEMENT -(f0,f1) HAS_TYPE cal_L_infty
  prod_cal_L_infty: JUDGEMENT *(f0,f1) HAS_TYPE cal_L_infty

  inf_norm(f):nnreal = essential_bound(f)                            % 6.6.5(1)

  inf_norm_scal: LEMMA inf_norm(c*f)    = abs(c)*inf_norm(f)         % 6.6.5(2)
  inf_norm_add:  LEMMA inf_norm(f0+f1) <= inf_norm(f0)+inf_norm(f1)  % 6.6.5(3)
  inf_norm_opp:  LEMMA inf_norm(-f)     = inf_norm(f)
  inf_norm_diff: LEMMA inf_norm(f0-f1) <= inf_norm(f0)+inf_norm(f1)
  inf_norm_eq_0: LEMMA inf_norm(f) = 0 <=> cal_N?(f)                 % 6.6.5(4)
  inf_norm_prod: LEMMA inf_norm(f0*f1) <= inf_norm(f0)*inf_norm(f1)  % 6.6.5(5)

  IMPORTING p_integrable_def[T,S,mu,1]

  g: VAR p_integrable

  holder_judge_inf_1: JUDGEMENT *(f,g) HAS_TYPE p_integrable
  holder_judge_inf_2: JUDGEMENT *(g,f) HAS_TYPE p_integrable

  holder_infty_1: LEMMA norm(f*g) <= inf_norm(f) * norm(g)

END cal_L_inf

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