If is normed, then (2.5).
Then we have the following assertions: (1) (T_{x_{0}}^{-1}) is normed.
In the case when the space is normed and the cone is normal, these results were proved in [9].
Indeed, if we suppose, to the contrary, that P is normal cone, then the topology τ* is normed (see, e.g.,[15]).
Hyers-Ulam-Rassias stability for the quadratic functional equation (1.3) was proved by Skof for functions, where is normed space and Banach space (see [25 28]).
In the case when the space is normed and under the additional assumption that the cone is normal, these results were firstly proved in [10].
A generalized Hyers-Ulam stability problem for the quadratic functional equation (1.1) was proved by Skof for mappings f : A → B, where A is normed space and B is a Banach space [8] (see [9 12]).
The space (P_{1}(I,E) ) is normed by Vert uVert _{P_{1}}=sup_{varphiin E^, Vert varphi Vert leq1} int_{1} ^{T}biglvert varphibigl u(x bigr) bigrvert,dlambda x, where λ stands for a Lebesgue measure on I.
Indeed, if this were the case, Theorem 1 would imply that the topology τ* is normed, which is impossible since an infinite dimensional space with the strongest locally convex topology cannot be metrizable (see, e.g., [14]).
For example, if a ∗ b = a b for a, b ∈ [ 0, 1 ], ( X, ∥ ⋅ ∥ ) is normed space and μ ( x, t ) = t t + ∥ x ∥. for all x, y, z ∈ X and t > 0. Then μ is a (standard) fuzzy normed and ( X, μ, ⋅ ) is a fuzzy normed space.
This version is normed and validated, and has shown similar validity with the original English RBANS form A in the local population.
