By Claudi Alsina
The e-book presents a complete evaluation of the characterizations of actual normed areas as internal product areas in response to norm derivatives and generalizations of the main simple geometrical houses of triangles in normed areas. because the visual appeal of Jordan-von Neumann's classical theorem (The Parallelogram legislations) in 1935, the sphere of characterizations of internal product areas has bought an important volume of recognition in numerous literature texts. furthermore, the innovations bobbing up within the thought of sensible equations have proven to be tremendous invaluable in fixing key difficulties within the characterizations of Banach areas as Hilbert areas. This ebook offers, in a transparent and specified variety, cutting-edge equipment of characterizing internal product areas by way of norm derivatives. It brings jointly effects which have been scattered in numerous courses during the last twenty years and comprises extra new fabric and strategies for fixing sensible equations in normed areas. hence the ebook can function a complicated undergraduate or graduate textual content in addition to a source e-book for researchers operating in geometry of Banach (Hilbert) areas or within the concept of useful equations (and their applications).
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Additional info for Norm Derivatives and Characterizations of Inner Product Spaces
1, one concludes that (vi) is equivalent to F (y, x)ρ′− (y, x) ≤ [F (y, x)]2 ≤ F (y, x)ρ′+ (y, x) in the case F (y, x) > 0, and also to the inequality with opposite signs in the other case. Both cases give finally ρ′− (y, x) ≤ F (y, x) ≤ ρ′+ (y, x). g. [Acz´el (1966); Kuczma (1985); Acz´el and Dhombres (1989)]). Its conditional form described below deserves further study. 1) where ⊥ denotes an orthogonal relation defined on X. For instance, in an inner product space (X, ·, · ) the functional X ∋ x → x, x ∈ R is orthogonally additive (Pythagoras theorem).
Now we turn our attention to the James orthogonality. 2 Let (X, · ) be a real normed linear space with dim X ≥ 2. s. whose inner product is F . Proof. Assume that F satisfies the above mentioned conditions. 3). 10) whence u + λv 2 − u 2λ 2 = u + λ 2 F (u,v) u 2 u−v 2 − u 2 2λ . So, taking limit when λ tends to zero from the right, we have ρ′+ (u, v) = ρ′+ u, 2 F (u, v) u−v u 2 =2 F (u, v) u u 2 2 − ρ′− (u, v) and therefore F (u, v) = (ρ′+ (u, v) + ρ′− (u, v))/2. 10) dividing by λ > 0 and taking limit when λ tends to infinity, one obtains v = v− ρ′+ (u, v) + ρ′− (u, v) u , u, v ∈ X, u = 0.
3) is satisfied with ⊥ := ⊥ρ . The question is: What about spaces which are not smooth? Assume that (X, · ) is a normed linear space with dim X ≥ 2. We will show that the relation ⊥ρ satisfies the four properties of the orthogonality space (see [Alsina, Sikorska and Tom´as (2007)]. The first three are easy to check. In order to check the fourth, we need some auxiliary results. 1 For any two vectors x and w in X, we have lim ρ′± (x + tw, w) = ρ′± (x + t0 w, w). t→t0 Proof. 2, we can write lim ρ′± (x + tw, w) = lim ρ′± (x + t0 w + sw, w) = ρ′± (x + t0 w, w).