Following Browder [27], we say that a Banach space has a weakly continuous duality mapping if there exists a gauge for which the duality mapping is single valued and continuous from the weak topology to the weak* topology, that is, for any with, the sequence converges weakly* to.
Following Browder [23], we say that a Banach space has a if there exists a gauge for which the duality mapping is single-valued and continuous from the weak topology to the weak* topology, that is, for any with, the sequence converges weakly* to.
Following Browder [22], we say that a Banach space has a weakly continuous duality map if there exists a gauge for which the duality map is single valued and weak-to-weak* sequentially continuous (i.e., if is a sequence in weakly convergent to a point, then the sequence converges weakly* to ).
Mr. Storr describes Mr. Richter's situation as a state of "fundamental alienation, of which art can be a finely calibrated gauge, and for which it is a consolation but not a remedy".
We say that a Banach space X has a weakly continuous duality map if there exists a gauge φ for which the duality map J φ is single-valued and weak-to-weak∗ sequentially continuous.
Associated to gauge φ is the duality map J φ : E → 2 E ∗ defined by J φ ( x ) = { x ∗ ∈ E ∗ : 〈 x, x ∗ 〉 = ∥ x ∥ φ ( ∥ x ∥ ), ∥ x ∗ ∥ = φ ( ∥ x ∥ ) }, ∀ x ∈ E. Following Browder [29], we say that a Banach space E has a weakly continuous duality map if there exists gauge φ for which the duality map J φ is single-valued and weak-to-weak∗ sequentially continuous.
Following Browder [7], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping J φ (x) is single-valued and continuous from the weak topology to the weak* topology, that is, for any {x n } with x n ⇀ x, the sequence {J φ (x n )} converges weakly* to J φ (x).
Following Browder [37], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping J φ ( x ) is single-valued and weak-to-weak∗ sequentially continuous (i.e., if { x n } is a sequence in E weakly convergent to a point x, then the sequence J φ ( x n ) converges weakly∗ to J φ ).
Recall that a Banach space E is said to have a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping J φ ( x ) is single-valued and weak-to-weak∗ sequentially continuous (i.e., if { x n } is a sequence in E weakly convergent to a point x, then the sequence J φ ( x n ) converges weakly∗ to J φ ).
Following Browder [1], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping J φ ( x ) is single-valued and weak-to-weak∗ sequentially continuous (i.e., if { x n } is a sequence in E weakly convergent to a point x, then the sequence J φ ( x n ) converges weakly∗ to J φ ).
