Throughout this paper, let X and Y be two real vector spaces and let φ : X × X × X × X → [0, ∞) be a function. For a mapping f : X × X → Y, consider the functional equation: f ( x + y, z + w ) + f ( x - y, z - w ) = 2 f ( x, z ) + 2 f ( y, w ). (1.1). The quadratic form f : ℝ × ℝ → ℝ given by f x, y) : = ax2 + bxy + cy2 is a solution of the Equation 1.1.
Let be real numbers, and and be two real vectors, then if and only if.
Let and be two real Hausdorff topological vector spaces, a nonempty, closed, and convex subset, and a closed, convex, and pointed cone with apex at the origin.
Throughout this paper, let X and Y be two real Hausdorff topological vector spaces and A a nonempty subset of X.
Let X and Z be two real Hausdorff topological vector spaces, K ⊂ X a nonempty closed convex subset and C ⊂ Z a closed convex cone.
In this paper, let X be a linear space, Y and Z be two real locally convex Hausdorff topological vector spaces with topological dual spaces (Y^) and (Z^), respectively.
For each i ∈ I, let X i and Y i be two real locally convex Hausdorff topological vector spaces and K i a nonempty convex compact subset of X i. Denote K i ˆ = ∏ j ∈ I, j ≠ i K j, K = ∏ i ∈ I K i = K i × K i ˆ, X = ∏ i ∈ I X i.
Let ((mathbb{K}, |cdot|_{mathbb{K}} )) and ((mathbb{H}, |cdot|_{mathbb{H}} )) be two real separable Hilbert spaces with their vector norms and inner products, respectively.
In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all.
In fact, they proved that a mapping between two real vector spaces and is a solution of (1.2) if and only if there exists a unique symmetric multi-additive mapping such that for all (see [7, 11]).
