  This work derives new results on strong consistent estimation and prediction
for autoregressive processes of order 1 in a separable Banach space B. The
consistency results are obtained for the component-wise estimator of the
autocorrelation operator in the norm of the space L(B) of bounded linear
operators on B. The strong consistency of the associated plug-in predictor then
follows in the B-norm. A Gelfand triple is defined through the Hilbert space
constructed in Kuelbs (1970)' lemma. A Hilbert--Schmidt embedding introduces
the Reproducing Kernel Hilbert space (RKHS), generated by the autocovariance
operator, into the Hilbert space conforming the Rigged Hilbert space structure.
This paper extends the work of Bosq (2000) and Labbas and Mourid 2002.
