In this work we prove the real Nullstellensatz for the ring
O
(
X
)
\mathcal {O}(X)
of analytic functions on a
C
C
-analytic set
X
⊂
R
n
X\subset \mathbb {R}^n
in terms of the saturation of Łojasiewicz’s radical in
O
(
X
)
\mathcal {O}(X)
: The ideal
I
(
Z
(
a
)
)
\mathcal {I}(\mathcal {Z}(\mathfrak {a}))
of the zero-set
Z
(
a
)
\mathcal {Z}(\mathfrak {a})
of an ideal
a
\mathfrak {a}
of
O
(
X
)
\mathcal {O}(X)
coincides with the saturation
a
L
~
\widetilde {\sqrt [L]{\mathfrak {a}}}
of Łojasiewicz’s radical
a
L
\sqrt [L]{\mathfrak {a}}
. If
Z
(
a
)
\mathcal {Z}(\mathfrak {a})
has ‘good properties’ concerning Hilbert’s 17th Problem, then
I
(
Z
(
a
)
)
=
a
r
~
\mathcal {I}(\mathcal {Z}(\mathfrak {a}))=\widetilde {\sqrt [\mathsf {r}]{\mathfrak {a}}}
where
a
r
\sqrt [\mathsf {r}]{\mathfrak {a}}
stands for the real radical of
a
\mathfrak {a}
. The same holds if we replace
a
r
\sqrt [\mathsf {r}]{\mathfrak {a}}
with the real-analytic radical
a
r
a
\sqrt [\mathsf {ra}]{\mathfrak {a}}
of
a
\mathfrak {a}
, which is a natural generalization of the real radical ideal in the
C
C
-analytic setting. We revisit the classical results concerning (Hilbert’s) Nullstellensatz in the framework of (complex) Stein spaces.
Let
a
\mathfrak {a}
be a saturated ideal of
O
(
R
n
)
\mathcal {O}(\mathbb {R}^n)
and
Y
R
n
Y_{\mathbb {R}^n}
the germ of the support of the coherent sheaf that extends
a
O
R
n
\mathfrak {a}\mathcal {O}_{\mathbb {R}^n}
to a suitable complex open neighborhood of
R
n
\mathbb {R}^n
. We study the relationship between a normal primary decomposition of
a
\mathfrak {a}
and the decomposition of
Y
R
n
Y_{\mathbb {R}^n}
as the union of its irreducible components. If
a
:=
p
\mathfrak {a}:=\mathfrak {p}
is prime, then
I
(
Z
(
p
)
)
=
p
\mathcal {I}(\mathcal {Z}(\mathfrak {p}))=\mathfrak {p}
if and only if the (complex) dimension of
Y
R
n
Y_{\mathbb {R}^n}
coincides with the (real) dimension of
Z
(
p
)
\mathcal {Z}(\mathfrak {p})
.