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

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

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

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

then reconsider f1 = f, h1 = h, g1 = g as VECTOR of (X *') by RLSUB_1:10;

then h1 = f1 + g1 by Th20b;

hence h = f + g by A1, RLSUB_1:13; :: thesis: verum

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

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

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

then reconsider f1 = f, h1 = h, g1 = g as VECTOR of (X *') by RLSUB_1:10;

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

assume
for x being Element of X holds h . x = (f . x) + (g . x)
; :: thesis: h = f + gassume A2:
h = f + g
; :: thesis: for x being Element of X holds h . x = (f . x) + (g . x)

let x be Element of X; :: thesis: h . x = (f . x) + (g . x)

h1 = f1 + g1 by A1, A2, RLSUB_1:13;

hence h . x = (f . x) + (g . x) by Th20b; :: thesis: verum

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

h1 = f1 + g1 by A1, A2, RLSUB_1:13;

hence h . x = (f . x) + (g . x) by Th20b; :: thesis: verum

then h1 = f1 + g1 by Th20b;

hence h = f + g by A1, RLSUB_1:13; :: thesis: verum