let X be RealNormSpace; :: thesis: for f, g, h being Point of (DualSp X) holds

( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

let f, g, h be Point of (DualSp X); :: thesis: ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

reconsider f1 = f, g1 = g, h1 = h as VECTOR of (R_VectorSpace_of_BoundedLinearFunctionals X) ;

( h = f + g iff h1 = f1 + g1 ) ;

hence ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) ) by Th24; :: thesis: verum

( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

let f, g, h be Point of (DualSp X); :: thesis: ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) )

reconsider f1 = f, g1 = g, h1 = h as VECTOR of (R_VectorSpace_of_BoundedLinearFunctionals X) ;

( h = f + g iff h1 = f1 + g1 ) ;

hence ( h = f + g iff for x being VECTOR of X holds h . x = (f . x) + (g . x) ) by Th24; :: thesis: verum