  Since the very beginning of the theory of linear logic it is known how to
represent the $\lambda$-calculus as linear logic proof nets. The two systems
however have different granularities, in particular proof nets have an explicit
notion of sharing---the exponentials---and a micro-step operational semantics,
while the $\lambda$-calculus has no sharing and a small-step operational
semantics. Here we show that the \emph{linear substitution calculus}, a simple
refinement of the $\lambda$-calculus with sharing, is isomorphic to proof nets
at the operational level.
  Nonetheless, two different terms with sharing can still have the same proof
nets representation---a further result is the characterisation of the equality
induced by proof nets over terms with sharing. Finally, such a detailed
analysis of the relationship between terms and proof nets, suggests a new,
abstract notion of proof net, based on rewriting considerations and not
necessarily of a graphical nature.
