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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 35, Issue 3, 2016, pp. 243–265**

**DOI: 10.4171/ZAA/1564**

Published online: 2016-09-13

Riesz-Like Bases in Rigged Hilbert Spaces

Giorgia Bellomonte^{[1]}and Camillo Trapani

^{[2]}(1) Università degli Studi di Palermo, Italy

(2) Università degli Studi di Palermo, Italy

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

*Keywords: *Riesz basis, rigged Hilbert spaces

Bellomonte Giorgia, Trapani Camillo: Riesz-Like Bases in Rigged Hilbert Spaces. *Z. Anal. Anwend.* 35 (2016), 243-265. doi: 10.4171/ZAA/1564