let X be non empty set ; :: thesis: for Y being ComplexNormSpace holds
( C_NormSpace_of_BoundedFunctions X,Y is reflexive & C_NormSpace_of_BoundedFunctions X,Y is discerning & C_NormSpace_of_BoundedFunctions X,Y is ComplexNormSpace-like )
let Y be ComplexNormSpace; :: thesis: ( C_NormSpace_of_BoundedFunctions X,Y is reflexive & C_NormSpace_of_BoundedFunctions X,Y is discerning & C_NormSpace_of_BoundedFunctions X,Y is ComplexNormSpace-like )
thus ||.(0. (C_NormSpace_of_BoundedFunctions X,Y)).|| = 0 by Th25; :: according to NORMSP_0:def 6 :: thesis: ( C_NormSpace_of_BoundedFunctions X,Y is discerning & C_NormSpace_of_BoundedFunctions X,Y is ComplexNormSpace-like )
for x, y being Point of (C_NormSpace_of_BoundedFunctions X,Y)
for c being Complex holds
( ( ||.x.|| = 0 implies x = 0. (C_NormSpace_of_BoundedFunctions X,Y) ) & ( x = 0. (C_NormSpace_of_BoundedFunctions X,Y) implies ||.x.|| = 0 ) & ||.(c * x).|| = |.c.| * ||.x.|| & ||.(x + y).|| <= ||.x.|| + ||.y.|| ) by Th25;
hence ( C_NormSpace_of_BoundedFunctions X,Y is discerning & C_NormSpace_of_BoundedFunctions X,Y is ComplexNormSpace-like ) by CLVECT_1:def 14, NORMSP_0:def 5; :: thesis: verum