Riesz-like bases in rigged Hilbert spaces

Abstract

The notions of Bessel sequence, Riesz-Fischer sequence and Riesz basis are generalized to a rigged Hilbert space D[t] ⊂ H ⊂ D^×[t^×]. A Riesz-like basis, in particular, is obtained by considering a sequence {ξ_n} ⊂ D which is mapped by a one-to-one continuous operator T : D[t] → H[\| \cdot \|] into an orthonormal basis of the central Hilbert space H of the triplet. The operator T is, in general, an unbounded operator in H. If T has a bounded inverse then the rigged Hilbert space is shown to be equivalent to a triplet of Hilbert spaces.