I have a question on the Moore-Aronszajn theorem regarding reproducing kernel Hilbert spaces. The theorem states that a reproducing kernel $K$ on a set $\Omega$ induces a RKHS. A kernel is defined as a function $K \colon \Omega \times \Omega \to \mathbb{C}$ which satisfies $K(z,w) = \overline{K(w, z)}$ and has a positive semidefinite Gram-like Matrix $(K(z_i, w_j))_{i,j=1}^{n}$, for any choice of $z_i, w_j \in \Omega$ and every $n \in \mathbb{N}$. According to Wikipedia and some other sources one can find when googling the theorem, the Hilbertspace is constructed in the following way:
Consider the space $E =$ span $\{ K(w, .) \colon w \in \Omega \}$ endowed with the inner product that arises from the Gram-like matrix mentioned above. The completion of this inner product space is then the RKHS with kernel $K$.
What baffles me about this construction is that it simply does not work i.m.o. since the inner product space defined in such a manner does not carry a positive definite inner product, as required for completion. The notation on this topic also is highly confusing. A positive semidefinite kernel as defined above is called positive definite in the literature.
thanks in advance.