  The celebrated Bishop theorem states that an operator is subnormal if and
only if it is the strong limit of a net (or a sequence) of normal operators. By
the Agler-Stankus theorem, $2$-isometries behave similarly to subnormal
operator in the sense that the role of subnormal operators is played by
$2$-isometries, while the role of normal operators is played by Brownian
unitaries. In this paper we give Bishop-like theorems for $2$-isometries. Two
methods are involved, the first of which goes back to Bishop's original idea
and the second refers to Conway and Hadwin's result of general nature. We also
investigate the strong and $*$-strong closedness of the class of Brownian
unitaries.
