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

for a being Real holds

( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

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

( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

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

( h = a * f iff h1 = a * f1 ) ;

hence ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) ) by Th25; :: thesis: verum

for a being Real holds

( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

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

( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

let a be Real; :: thesis: ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) )

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

( h = a * f iff h1 = a * f1 ) ;

hence ( h = a * f iff for x being VECTOR of X holds h . x = a * (f . x) ) by Th25; :: thesis: verum