5. Construction of elements of a banach spaces from sub-spaces

  • Tomokkazu Onzukah
Keywords: Banach spaces, sub-spaces, constraints, inequality constraint


This research proposes two important developments on the theorems of S. N. Bern- stein for Banach space. This paper proves that if X is the arbitrarily infinite-dimension Banach spaces, Yn is the sequence of strict nest sub-spaces of X and if dn is the non-increase sequences of non-negative number tends to 0, then for any c ∈ (0, 1] it could be found xc ∈ X, so that the distances ρ(xc, Yn) from xc to Yn satisfy cdn ≤ ρ(xc, Yn) ≤ 4cdn, for all n ∈ N. The above inequality equation is proved by firstly refining Borodin’s method for Banach space by fading the conditions on the sequences {dn}. The faded conditions on dn require improvement of Borodin constructions to get the elements in X, whose distance from the nest subspace is exactly the provided value dn.