  We consider nonconforming methods for symmetric elliptic problems and
characterize their quasi-optimality in terms of suitable notions of stability
and consistency. The quasi-optimality constant is determined and the possible
impact of nonconformity on its size is quantified by means of two alternative
consistency measures. Identifying the structure of quasi-optimal methods, we
show that their construction reduces to the choice of suitable linear operators
mapping discrete functions to conforming ones. Such smoothing operators are
devised in the forthcoming parts of this work for various finite element
spaces.
