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Figure 1. Stephan Banach 1892-1945
Definition 1.1. Two norms |.| and . on a Banach space X are called equivalent if there exist positive constants A, B with A x ≤ |x|≤ B x for all x ∈ X. If X is finite dimensional all norms on X are equivalent. Through this paper X is a real Banach space with norm . and dual X∗, S(X) and B(X) are, respectively, the unit sphere and the closed unit ball of X. In addition,
∞ p ∞ ∞ ℓ , ℓ denote the space of all real sequence (xn)n=1 such that (xn)n=1 is bounded, ∞ p and |xn| < ∞, respectively. The reader is referred to [23] for undefined terms nX=1 and notation.
2. Differentiable norms and renorming theory.
Differentiability of functions on Banach spaces is a natural extension of the notion of a directional derivative on Rn. A function f : X → R is said to be Gateaux differentiable at x ∈ X if there exists a functional A ∈ X∗ such that f(x + ty) − f(x) A(y) = lim , for all y ∈ X. In this case, A is called Gateaux t→0 t derivative of f. If the above limit exists uniformly for each y ∈ S(X), then f is called Fr´echet differentiable at x with Fr´echet derivative A. Similarly, the norm . of X is Gateaux or Fr´echet differentiable at a non-zero x ∈ X if the function f(x)= x has the same property at x. Two important more strong notions of differentiability are obtained as uniform versions of both Fr´echet and Gateaux differentiability. x + ty − x The norm . of X is uniformly Fr´echet differentiable if lim exists t→0 t uniformly for (x, y) ∈ S(X) × S(X). Also, it is uniformly Gateaux differentiable if EQUIVALENT NORMS 3