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