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 Th22; :: 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 Th22;

hence ( C_NormSpace_of_BoundedFunctions (X,Y) is discerning & C_NormSpace_of_BoundedFunctions (X,Y) is ComplexNormSpace-like ) by CLVECT_1:def 13; :: thesis: verum

( 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 Th22; :: 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 Th22;

hence ( C_NormSpace_of_BoundedFunctions (X,Y) is discerning & C_NormSpace_of_BoundedFunctions (X,Y) is ComplexNormSpace-like ) by CLVECT_1:def 13; :: thesis: verum