CONDITIONS OF DENSITY OF THE CANONICAL EMBEDDING OF BANACH SPACE Х IN Е* IN THE TERMS OF BORDER
The symbol Е ∈ (TD) - (abbreviated from "total" and "reach") is used by us in case when Е is strongly closed total subspace in X*, and all its elements attain their norms on the unit ball B (X) of Banach space X. The condition Е ∈ (TD) is necessary for canonical isometry of X and Е*, i. e. established between X and Х* duality, but it is not a sufficient condition. Moreover, as it has been shown earlier, X can canonically densely on norm be embedded in Е*. Let - canonical embedding. The main results of this paper are the following: Theorem 1. Let X - Banach space, Е ∈ (TD). Then the image j (X) is everywhere dense in strong topology, if Е* coincides with strong closure of linear span of any border of ball В(Е*). Theorem 2. Let E - Banach space. If there is such a border Г ⊂ В(Е*), that strong closure of its linear span does not coincide with Е* will be such Banach space X, that Е is isometric for subspace in X*, Е ∈ (TD) rather X, but j (X) is not dense in Е*.