The norm is said to be uniformly Gâteaux differentiable if for y ∈ S, the limit is attained uniformly for x ∈ S. A Banach space E is said to be uniformly convex if for all ε ∈ [ 0, 2 ], there exists δ ε > 0 such that ∥ x ∥ = ∥ y ∥ = 1 implies ∥ x + y ∥ 2 < 1 − δ ε whenever ∥ x − y ∥ ≥ ε.
The norm of is a uniformly differentiable if for each, the limit is attained uniformly for.
We say that g is uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for (xin C) and (|y|=1).
Finally, f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x ∈ C and ∥ y ∥ = 1.
We say that f is uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x ∈ C and ∥ y ∥ = 1.
Finally, f is said to be uniformly Fréchet differentiable on a subset D of X, if the limit is attained uniformly, for x ∈ D and ∥ y ∥ = 1.
Finally, f is called uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for (xin C) and (|y|=1).
The function f called Fréchet differentiable at x if the limit in (1.2) is attained uniformly for all y ∈ E such that ∥ y ∥ = 1 and f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x ∈ C and ∥ y ∥ = 1.
The norm of X is said to be Fréchet differentiable if, for each x ∈ U, this limit is attained uniformly for all y ∈ U. A function ρ : [ 0, ∞ ) → [ 0, ∞ ) defined by ρ = sup { 1 2 ( ∥ x + y ∥ + ∥ x − y ∥ ) − 1 : x, y ∈ X, ∥ x ∥ = 1, ∥ y ∥ = τ }. is called the modulus of smoothness of X.
The space E is said to have a uniformly Gâteaux differentiable norm, if for each y ∈ U, the limit is attained uniformly in x ∈ U.
X is said to have a uniformly Gâteaux differentiable norm if for each y ∈ U, the limit is attained uniformly for x ∈ U.
