Sentence examples for random norms from inspiring English sources

Hence, one needs to discuss on a new family of random norms.

Finally, a type of Gâteaux differentiability is defined for random norms, and its relation to random smoothness is given.

For random norms we present the following definition of Gâteaux differentiability, which is equivalent to that in [19], Definition 5.2.

Finally, a type of Gâteaux differentiability equivalent to that in [19] is introduced for random norms, and its relation to random smoothness is given.

Under Definition 4.3 we can establish the relations among supporting functionals, points of random smoothness and Gâteaux differentiability of random norms.

The development of RN spaces in the direction of functional analysis led Guo to present a new version of RM and RN spaces in [2], where the random distances or random norms are defined to be the equivalence classes of nonnegative random variables according to the new versions.

In the last two decades, several form of mixed type functional equation and its Ulam Hyers stability are dealt in various spaces like Fuzzy normed spaces, Random normed spaces, Quasi Banach spaces, Quasinormed linear spaces and Banach algebra by various authors like [31 40].

Since a random locally convex module is a generalization of a random normed module and a random norm is ({L}^{0} -convex, it is easy to see that a random norm is also a proper local function.

The random distance between two points in an original random metric space (briefly, an RM space) is a nonnegative random variable defined on some probability space, similarly, the random norm of a vector in an original random normed space (briefly, an RN space) is a nonnegative random variable defined on some probability space.

Section 4 is focused on the Gâteaux differentiability of random norm, where some inequality techniques are employed in combination with stratification analysis in random normed modules to derive the main result Theorem 4.6.

(RNM-1) ∥ ξ x ∥ = | ξ | ∥ x ∥, ∀ ξ ∈ L 0 ( F, K ) and x ∈ E. Then ( E, ∥ ⋅ ∥ ) is called a random normed module (briefly, an RN module) over K with base ( Ω, F, P ), the random norm ∥ ⋅ ∥ with the property (RNM-1) is also called an L 0 -norm on E (a mapping only satisfying (RN-3) and (RNM-1) above is called an L 0 -seminorm on E).