  The notion of duality is a key element in understanding the interplay between
the stability and chromatic numbers of a graph. This notion is a central aspect
in the celebrated theory of perfect graphs, and is further and deeply developed
in the context of the Lov\'asz theta function and its equivalent
characterizations and variants. The main achievement of this paper is the
introduction of a new family of norms, providing upper bounds for the stability
number, that are obtained from duality from the norms motivated by Hoffman's
lower bound for the chromatic number and which achieve the (complementary)
Lov\'asz theta function at their optimum. As a consequence, our norms make it
formal that Hoffman's bound for the chromatic number and the Delsarte-Hoffman
ratio bound for the stability number are indeed dual. Further, we show that our
new bounds strengthen the convex quadratic bounds for the stability number
studied by Luz and Schrijver, and which achieve the Lov\'asz theta function at
their optimum. One of the key observations regarding weighted versions of these
bounds is that, for any upper bound for the stability number of a graph which
is a positive definite monotone gauge function, its gauge dual is a lower bound
on the fractional chromatic number, and conversely. Our presentation is
elementary and accessible to a wide audience.
