Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, Tehran, Iran.
In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new denition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we dene the fusion biorthogonal sequence, Bessel fusion basis, Hilbert fusion basis and obtain some character-izations of them. we study orthonormal fusion systems and Riesz fusion bases for Hilbert spaces. we consider the stability of fusion bases under small perturbations. We also general-ized a result of Paley-Wiener  to the situation of fusion basis.