Sentence examples for random norm from inspiring English sources

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).

In this case, P μ, ν is called an intuitionistic random norm.

In this case, Λ μ, ν is called an intuitionistic random norm.

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].

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.