By Hardy G. H.

Hardy's natural arithmetic has been a vintage textbook for the reason that its e-book in1908. This reissue will deliver it to the eye of a complete new new release of mathematicians.

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We refer to Megginson [85, p. 10. 2 Banach sequence spaces Let (X, · ) be a normed linear space. For a fixed positive integer n, consider the Cartesian power of X, Xn = X × · · · × X = {x = (x1 , . . , xn ) ∈ Xn : xi ∈ X}. Under the usual addition and scalar multiplication, it becomes a normed space when equipped with any of the following p-norms: x p ( x p + · · · + x p )1/p , 1 ≤ p < ∞; 1 n = max{ x , . . , x }, p = ∞, 1 n for all x = (x1 , . . , xn ) ∈ Xn . Note that these spaces are the vector-valued analogues of the pn spaces (cf.

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]. Dragomir mentioned some other types of semi-inner product which were considered by other mathematicians such as Miliˇci´c, Tapia, Pavel and Dincˇa [37]. In a normed linear space (X, · ), the mapping f : X → R defined by f (x) = 12 x 2 is convex and the following limits exist x, y i = lim− t→0 y + tx 2 2t − y 2 and x, y s = lim+ t→0 y + tx 2 2t − y 2 , for any x, y ∈ X [37, 115].

For any (x1 , . . , xn ) ∈ Rn , the mapping · p : Rn → R defined by (x1 , . . , xn ) p (|x1 |p + · · · + |xn |p )1/p , 1 ≤ p < ∞; = max{|x |, . . , |x |}, p=∞ 1 n is a norm. The space Rn is a Banach space with the norm p n. · p ; and it is denoted by From the last two examples, it is important to note that we may equip a vector space with more than one norm. When two norms in a vector space induce the same topology, they are said to be equivalent. It is also important to note if · and | · | are two norms on a vector space X, then they are equivalent if and only if there exist positive constants c1 and c2 such that c1 | x | ≤ x ≤ c2 | x |, for all x ∈ X.