ENTIRE CURVES HAVING BOUNDED l-INDEX IN ℓ∞.

In this paper we propose an approach to introduce a concept of bounded index in an infinite-dimensional space. Our object of investigation is the space ℓ^∞ equipped with the norm ∥x∥_∞ = sup{|x_n|: n ∈ N}. We consider entire curves from C to ℓ_∞ and prove a proposition indicating connection between of the l-index boundedness of every component of the curve and the l-index boundedness of the curve. Moreover, we obtain sufficient conditions of the l-index boundedness of entire curves in the space. They describe local behavior of norm of derivatives of the entire curves on the discs. Also, there is posed a problem on necessary conditions of the l-index boundedness of entire curves in infinite-dimensional spaces. [ABSTRACT FROM AUTHOR]