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By Pogorelov, A.V.

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**Example text**

1. Let 1 < p < ∞. The space Lp ([0, 1], X) is a smooth (Fr´echet smooth) Banach space whenever X is smooth (Fr´echet smooth, respectively). We refer to McShane [84, p. 1. 2. Let 1 < p < ∞. The space Lp ([0, 1], X) is a reflexive Banach space if X is reflexive. Bochner in [13, p. 930] stated that if X and its dual X∗ are of (D)-property (namely, any function of bounded variation is differentiable almost everywhere [13, p. 914–915]) and X is reflexive, then Lp ([0, 1], X) is reflexive. However, further studies have shown that these conditions could be reduced to a simpler condition.

The p-th power mean of f on [a, b], which is defined by [p] M[a,b] (f ) = 1 b−a 1 p b p f (x) dx , a [r] [s] is increasing on R, that is, if −∞ ≤ r < s ≤ ∞, then, M[a,b] (f ) ≤ M[a,b] (f ). 5, we obtain the following consequence. 6 (Kikianty and Dragomir [71]). The p-HH-norm is monotonically increasing as a function of p on [1, ∞], that is, for any 1 ≤ r < s ≤ ∞ and (x, y) ∈ X2 , we have (x, y) r−HH ≤ (x, y) s−HH . Proof. Consider the nonnegative function f (t) = (1 − t)x + ty on [0, 1]. 3), we conclude that f ∈ Lp [0, 1] for 1 ≤ p ≤ ∞.

X, λy] = λ[x, y] for all x, y ∈ X and λ a scalar in K and λ is the conjugate of λ. A vector space equipped with a semi-inner product is called a semi-inner product space. According to Lumer [54], the importance of this concept is that every normed space can be represented as a semi-inner product space, so that the theory of operators on Banach spaces may be penetrated by Hilbert spaces type arguments. As it has more general axioms, obviously there are some limitations on the theory of semi-inner product spaces in comparison to that of Hilbert spaces [54].