A Reverse Theorem on the ·-w* Continuity of the Dual Map. (10th March 2015) - de Kock, Mienie; García-Pacheco, Francisco Javier
Abstract : Given a Banach spaceX, x ∈ 𝖲 X, and𝖩 X x = x * ∈ 𝖲 X * : x * x = 1, we define the set𝖩 X * x of allx * ∈ 𝖲 X * for which there exist two sequencesx n n ∈ N ⊆ 𝖲 X ∖ { x } andx n * n ∈ N ⊆ 𝖲 X * such thatx n n ∈ N converges tox, x n * n ∈ N has a subnetw * -convergent tox *, andx n * x n = 1 for alln ∈ N . We prove that ifX is separable and reflexive andX * enjoys the Radon-Riesz property, then𝖩 X * x is contained in the boundary of𝖩 X x relative to𝖲 X * . We also show that ifX is infinite dimensional and separable, then there exists an equivalent norm onX such that the interior of𝖩 X x relative to𝖲 X * is contained in𝖩 X * x .