  Classical conditions for ensuring the robust stability of a system in
feedback with a nonlinearity include passivity, small gain, circle, and
conicity theorems. We present a generalized and unified version of these
results in an arbitrary semi-inner product space, which avoids many of the
technicalities that arise when working in traditional extended spaces. Our
general formulation clarifies when the sufficient conditions for robust
stability are also necessary, and we show how to construct worst-case scenarios
when the sufficient conditions fail to hold. Finally, we show how our general
result can be specialized to recover a wide variety of existing results, and
explain how properties such as boundedness, causality, linearity, and
time-invariance emerge as a natural consequence.
