An Application of the Generalised James' Weak Compactness Theorem

We provide a short proof of following theorem, due to Delbaen and Orihuela and independently, Pérez-Aros and Thibault. Let

be a nonempty closed and bounded convex subset of a Banach space

(X,\|\cdot\|) ( X , ∥ ⋅ ∥ )

and let

W W

be a nonempty weakly compact subset of

(X, \|\cdot\|) ( X , ∥ ⋅ ∥ )

. If we have

x_0^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0\}\ \ \ \text{and}\ \ \ \mathrm{argmax}(y^*|_A) \not= \varnothing x 0 ∗ ∈ { x ∗ ∈ X ∗ : sup ⁡ a ∈ A x ∗ ( a ) < 0 }    and    a r g m a x ( y ∗ ∣ A ) ≠ ∅

for each

y^* \in \{x^* \in X^*: \sup_{a \in A} x^*(a) <0 y ∗ ∈ { x ∗ ∈ X ∗ : sup ⁡ a ∈ A x ∗ ( a ) < 0

and

\sup_{w \in W} |(x^*-x^*_0)(w)|<1\} sup ⁡ w ∈ W ∣ ( x ∗ − x 0 ∗ ) ( w ) ∣ < 1 }

, then

A A

is weakly compact.