Compactness, Optimality, and Risk

One-sided James' compactness theorem

“…Remark that this last proposition together with the unbounded version of Simons' inequality [4,Theorem 10.5] gives an alternative proof of Theorem 6.…”

“…so lim sup n c * , x n ≤ α + ε for each c * ∈ C. This will be a consequence of the unbounded version of Simons' inequality (see [4,Theorem 10.5]) once we have checked that every x ∈ co σp {x n : n ≥ 1} attains its supremum on C. Fix such an x and notice that x ∈ L D . Assume that x does not attain its (finite) supremum α on C, so there is a sequence of elements (x * n ) n∈N in C satisfying α − 1 ≤ x 0 , x * n ≤ α and x * n ≥ n for each n ∈ N. By the Uniform Boundedness Principle there exists z ∈ S E such that ( z, x * n ) n∈N is unbounded above.…”

“…This will be a consequence of the unbounded version of Simons' inequality (see [4,Theorem 10.5]) once we have checked that every x ∈ co σp {x n : n ≥ 1} attains its supremum on C. Fix such an x and notice that x ∈ L D . Assume that x does not attain its (finite) supremum α on C, so there is a sequence of elements…”