Entire curves having bounded $l$-index in $\ell_{\infty}$

A. I. Bandura
Department of Advanced Mathematics Ivano-Frankivsk National Technical University of Oil and Gas Ivano-Frankivsk, Ukraine
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 $\ell^\infty$ equipped with the norm $\|x\|_{\infty }=\sup\{|x_{n}|\colon n\in\mathbb{N}\}.$ We consider entire curves from $\mathbb{C}$ to ${\ell}_{\infty}$ and prove 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 boundednes 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.
bounded index; bounded $l$-index; entire curve; $\ell_{\infty}$
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