Conditional convergence of rows

U ovom radu opisuju se pojedina svojstva uvjetno konvergentnih redova u različitim prostorima. U prvom poglavlju bavimo se redovima realnih brojeva te dokazujemo Riemannov teorem, koji pokazuje na koje se načine može promijeniti suma uvjetno konvergentnog reda u R mijenjanjem poretka članova reda. U drugom poglavlju bavimo se redovima vektora u Banachovim prostorima. Na početku poglavlja diskutiraju se odnosi obične, apsolutne i bezuvjetne konvergencije u različitim prostorima. Nadalje, definira se domena sume reda te se dokazuje Steinitzov teorem, koji daje opis domene sume uvjetno konvergentnog reda u konačnodimenzionalnim Banachovim prostorima. Na kraju se diskutira slučaj beskonačnodimenzionalnog prostora i pokazuje se da uvođenjem dodatnih uvjeta Steinitzov teorem vrijedi i u prostorima \(L_p\) , gdje je 1 < p < \(\infty\).In this paper we describe certain properties of conditionally convergent series in different spaces. In the first chapter we deal with series of real numbers and we prove the Riemann series theorem, which shows the ways in which the sum of a conditionally convergent series can be changed by reordering the terms of the series. In the second chapter we deal with series of vectors in Banach spaces. At the beginning of the chapter we discuss the relations between regular, absolute, and unconditional convergence. Furthermore, we define the domain of sums of a series and prove Steinitz’s theorem, which gives a description of the domain of sums of conditionally convergent series in finite-dimensional Banach spaces. In the end we discuss the case of an infinite-dimensional space and we show that, by adding additional conditions to Steinitz’s theorem, it holds in spaces \(L_p\) , where 1 < p < \(\infty\