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We prove a linear version of Dawson-G{\"a}rtner theorem: weak large deviationprinciples and the equality --s = p* between the negentropy and theFenchel-Legendre transform of the pressure are preserved through linearprojective limits. As a result, the equality --s = p* holds in great generalityfor empirical means of independent and identically distributed random variables(Cram{\'e}r's theory), e.g. in any measurable normed space, and even in anyprojective limit of such spaces. Eventually, we give an original example where--s is different from p* and discuss the dual equality.

A linear version of Dawson-G{\"a}rtner theorem -- Applications to Cram{\'e}r's theory