Norm Derivatives and Characterizations of Inner Product Spaces

On the Hyperstability of a Pexiderised -Quadratic Functional Equation on Semigroups

“…Iz. EL-Fassi and J. Brzdęk [2] All solutions f : G → E to (1.2) have been described in [25], in the situation when G is commutative and E is an abelian group that is uniquely divisible by 2 (that is, for every x ∈ E there is a unique y ∈ E with x = 2y); each of them has the form f (x) ≡ L(x, x) + a(x) with some homomorphism a : G → E and a symmetric L : G 2 → E such that a(σ(x)) = a(x), x ∈ G, (1.4) L(xz, y) = L(x, y) + L(z, y), x, y, z ∈ G, (1.5) L(σ(x), y) = −L(x, y), x, y ∈ G.…”

On a functional equation characterizing linear similarities

“…Observe that the above condition implies that f | span{y} (•) is bounded below on the segment {γy : γ ∈ [1,2]}. The proof of Theorem 5 is complete.…”

“…Now we define ρ + -orthogonality: x⊥ ρ + y :⇔ ρ + (x, y) = 0. The following properties can be found, e.g., in [1,2].…”

“…If X is smooth, then the following condition holds (see [1]): (nd6) for any two-dimensional subspace P of X and for every x ∈ P , λ ∈ (0, +∞), there exists a y ∈ P such that x⊥ ρ + y and x+y⊥ ρ + λx−y. In a real inner product space (X, •|• ), given the triangle determined by two linearly independent vectors x, y and the zero vector (i.e., {x, y, 0}), one can compute the height vector from y to the side x and orthogonal to x using the formula h(x, y):=y− x|y x 2 x.…”

Characterizations of Rotundity and Smoothness by Approximate Orthogonalities

“…Convexity of the norm yields that the above definitions are meaningful. Now, we recall their useful properties (the proofs can be found in [1] and [4]):…”