let X be non empty set ; :: thesis: for S being SigmaField of X
for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space M,k) st f in x holds
( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) )
let S be SigmaField of X; :: thesis: for M being sigma_Measure of S
for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space M,k) st f in x holds
( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) )
let M be sigma_Measure of S; :: thesis: for f being PartFunc of X,REAL
for k being positive Real
for x being Point of (Lp-Space M,k) st f in x holds
( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) )
let f be PartFunc of X,REAL ; :: thesis: for k being positive Real
for x being Point of (Lp-Space M,k) st f in x holds
( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) )
let k be positive Real; :: thesis: for x being Point of (Lp-Space M,k) st f in x holds
( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) )
let x be Point of (Lp-Space M,k); :: thesis: ( f in x implies ( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) ) )
assume A1: f in x ; :: thesis: ( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) )
x in the carrier of (Pre-Lp-Space M,k) ;
then x in CosetSet M,k by VSPDef6X;
then consider g being PartFunc of X,REAL such that
A2: ( x = a.e-eq-class_Lp g,M,k & g in Lp_Functions M,k ) ;
g in x by A2, EQC01;
then ( f a.e.= g,M & f in Lp_Functions M,k & g in Lp_Functions M,k ) by A1, Lm10;
hence ( x = a.e-eq-class_Lp f,M,k & ex r being Real st
( 0 <= r & r = Integral M,((abs f) to_power k) & ||.x.|| = r to_power (1 / k) ) ) by Lm17x, A1, A2, EQC02bx; :: thesis: verum