  We study $\ell^1$ regularized least squares optimization problem in a
separable Hilbert space. We show that the iterative soft-thresholding algorithm
(ISTA) converges linearly, without making any assumption on the linear operator
into play or on the problem. The result is obtained combining two key concepts:
the notion of extended support, a finite set containing the support, and the
notion of conditioning over finite dimensional sets. We prove that ISTA
identifies the solution extended support after a finite number of iterations,
and we derive linear convergence from the conditioning property, which is
always satisfied for $\ell^1$ regularized least squares problems. Our analysis
extends to the the entire class of thresholding gradient algorithms, for which
we provide a conceptually new proof of strong convergence, as well as
convergence rates.
