let X be RealNormSpace-Sequence; :: thesis: for Y being RealBanachSpace holds R_NormSpace_of_BoundedMultilinearOperators (X,Y) is RealBanachSpace

let Y be RealBanachSpace; :: thesis: R_NormSpace_of_BoundedMultilinearOperators (X,Y) is RealBanachSpace

for seq being sequence of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)) st seq is Cauchy_sequence_by_Norm holds

seq is convergent by Th42;

hence R_NormSpace_of_BoundedMultilinearOperators (X,Y) is RealBanachSpace by LOPBAN_1:def 15; :: thesis: verum

let Y be RealBanachSpace; :: thesis: R_NormSpace_of_BoundedMultilinearOperators (X,Y) is RealBanachSpace

for seq being sequence of (R_NormSpace_of_BoundedMultilinearOperators (X,Y)) st seq is Cauchy_sequence_by_Norm holds

seq is convergent by Th42;

hence R_NormSpace_of_BoundedMultilinearOperators (X,Y) is RealBanachSpace by LOPBAN_1:def 15; :: thesis: verum