BANACHSTEINHA US TYPE THEOREM IN LOCALLY CONVEX SPACES FOR æ-LOCALLY LIPSCHITZIAN CONVEX PROCESSES

Lipschitzian if ∀C ∈ σ, ∀β ∈ I : L(β,C) ≡ L(qβ , C)(T ) < +∞. By Lip(Xλ, Yμ, σ) we denote the vector space of σ-locally Lipschitzian operators. Note that Lip(Xλ, Yμ, σ) is a locally convex space under the locally convex topology τ(λ, μ, σ) generated by the family of semi norms L(β,C), β ∈ I, C ∈ σ. The operator T : (X,λ)→ (Y, μ) is said to be sequentially continuous if for every sequence (xn) of X and every x ∈ X such that xn λ −→ x one has Txn μ −→ Tx. T is said to be bounded if T sends bounded sets in (X,λ) into bounded sets in (Y, μ). Clearly, continuous operators are sequentially continuous, sequentially continuous operators are bounded, and linear bounded operators are σ-locally Lipschitzian; but in general, converse implications fail. Let X ′, Xs, Xb and XL σ denote respectively the family of continuous linear functionals sequentially continuous linear functionals, linear bounded functionals and σ-locally Lipschitzian functionals on (X,λ). In general, the inclusions X ′ ⊂ Xs ⊂ b ⊂ σ are strict. Let θ(X,XL σ ) denote the topology of uniform convergence on σ(X L σ , X)-Cauchy sequences of XL σ . Note that if σ = β(Xλ), then X b = XL σ and consequently, θ(X,X L σ ) = θ(X,X b). It has been show in [1] that, if Tn : (X,λ) → (Y, μ), n ∈ N is a sequence of σ-locally Lipschitzian operators admitting for each x ∈ X a weak limit limTnx = Tx, then the limit operator T maps θ(X,XL σ )-bounded sets into bounded sets. Our objective in this paper is to generalize the above result to σ-locally Lipschitzian convex processes.