Metric <span lang="EN-US">spaces are specific types of topological spaces with pleasing “geometric” characteristics and they have a number of appealing properties and are commonly used in both pure and applied sciences. In this work, the structure of cartesian product space in the setting of a fuzzy b-metric space (Fb-M space) framework is introduced, which is an extension allows to create the large-scale structure for the space of the type fuzzy b-metric. The possibility of transferring some of the results and important features related to Fb-M space to this suggested space are discussed and demonstrated. The Cartesian product of two Fb-M spaces is proved as Fb-M space, this allows, to investigate and prove the topological expectation on the fuzzy product b-metric space. Furthermore, certain specific fuzzy b-metrics on F², and fuzzy Euclidean plane are obtained in this way.</span>

In this paper, depending on the notion of fuzzy length space we define the Cartesian product of two fuzzy length spaces. we proved that the Cartesian product of two fuzzy length spaces is a fuzzy length space. More accurately, the Cartesian product of two complete fuzzy length spaces is proved to be a complete fuzzy length space. Furthermore, the definitions of sequentially fuzzy compact fuzzy length space, countably fuzzy compact fuzzy length space, locally fuzzy compact fuzzy length space are introduced, and theorems related to them are proved.

In this paper, an orthonormal family has been constracted{ } of polynomials of degree six is first constructed by using Gram-Schmidt orthonormalization process on Bernstein polynomials { } .Then, an orthonormal Bernstein operational matrix of derivative is derived.Finally, the orthonormal Bernstein expansions along with operational matrix of derivative are applied for variational problemsapproximately.