let X be RealNormSpace; :: thesis: for f, h being VECTOR of (R_VectorSpace_of_BoundedLinearFunctionals 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 VECTOR of (R_VectorSpace_of_BoundedLinearFunctionals 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) )

A1: R_VectorSpace_of_BoundedLinearFunctionals X is Subspace of X *' by Th22, RSSPACE:11;

then reconsider f1 = f, h1 = h as VECTOR of (X *') by RLSUB_1:10;

then h1 = a * f1 by Th21b;

hence h = a * f by A1, RLSUB_1:14; :: 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 VECTOR of (R_VectorSpace_of_BoundedLinearFunctionals 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) )

A1: R_VectorSpace_of_BoundedLinearFunctionals X is Subspace of X *' by Th22, RSSPACE:11;

then reconsider f1 = f, h1 = h as VECTOR of (X *') by RLSUB_1:10;

hereby :: thesis: ( ( for x being VECTOR of X holds h . x = a * (f . x) ) implies h = a * f )

assume
for x being Element of X holds h . x = a * (f . x)
; :: thesis: h = a * fassume A2:
h = a * f
; :: thesis: for x being Element of X holds h . x = a * (f . x)

let x be Element of X; :: thesis: h . x = a * (f . x)

h1 = a * f1 by A1, A2, RLSUB_1:14;

hence h . x = a * (f . x) by Th21b; :: thesis: verum

end;let x be Element of X; :: thesis: h . x = a * (f . x)

h1 = a * f1 by A1, A2, RLSUB_1:14;

hence h . x = a * (f . x) by Th21b; :: thesis: verum

then h1 = a * f1 by Th21b;

hence h = a * f by A1, RLSUB_1:14; :: thesis: verum