Sentence examples for be a random normed from inspiring English sources

Let ((E,mu,ast)) be a random normed space.

Definition 1.5 Let ( X, μ, T ) be a random normed space.

Let S be a random normed algebra with identity e, and let f be an L0-function on S satisfying f e) = 1 and f x2) = f(x 2 for all x ∈ S. Then f is multiplicative.

Let S be a random normed algebra, A ∈ ℱ and f be an L0-linear function on S, i.e., a mapping from S to L 0 ( ℱ, C ) such that f(ξ · x + η · y) = ξf(x) + ηf y) for all ξ, η ∈ L 0 ( ℱ, C ) and x, y ∈ S. Then f is called multiplicative if f xy) = f(x f y) for all x, y ∈ S and is called nonzero if there exists x ∈ S such that [ f ( x ) ≠ 0 ] = Ω ̃.

Then is a random normed space, where is the minimum -norm.

Throughout this paper, assume that ( X, μ, T M ) is a random normed algebra and that ( Y, μ, T M ) is a complete random normed algebra.

It is easy to think that S is a random metric on E, of course, ((E,S)) is a random normed space.

Definition 1.3 A random normed algebra is a random normed space with algebraic structure such that ( R N 4 ) μ x y ( t s ) ≥ μ x ( t ) μ y ( s ) for all x, y ∈ X and all t, s > 0. Example 1.4 Every normed algebra ( X, ∥ ⋅ ∥ ) defines a random normed algebra ( X, μ, T M ), where μ x ( t ) = t t + ∥ x ∥. for all t > 0. This space is called the induced random normed algebra.

A random normed algebra ((X,mu,T,T')) is a random normed space ((X,mu,T)) with algebraic structure such that (RN4) (mu_{xy} ts)geq T'(mu_{x}(t), mu_{y}(s))) for all (x,yin X) and (t,s> 0), in which (T') is a continuous t-norm.  .

Lemma 1 Let ( X, F, ∗ ) be a random 2-normed space.

Theorem 1 Let ( X, F, ∗ ) be a random 2-normed space.