In the first two chapters we review the theory developped by
Cartan, Whitney and Tognoli. Then Nullstellensatz is proved both
for Stein algebras and for the algebra of real analytic functions
on a C-analytic space. Here we find a relation between real
Nullstellensatz and seventeenth Hilbert's problem for positive
semidefinite analytic functions. Namely, a positive answer to
Hilbert's problem implies a solution for the real Nullstellensatz
more similar to the one for real polinomials. A chapter is devoted
to the state of the art on this problem that is far from a complete
answer. In the last chapter we deal with inequalities. We describe
a class of semianalytic sets defined by countably many global real
analytic functions that is stable under topological properties and
under proper holomorphic maps between Stein spaces, that is,
verifies a direct image theorem. A smaller class admits also a
decomposition into irreducible components as it happens for
semialgebraic sets. During the redaction some proofs have been
simplified with respect to the original ones.
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