Consider two identical particles $A$ and $B$. The combined Hilbert space $\mathcal{H}_A\otimes \mathcal{H}_B \newcommand{\ket}[1]{\left|#1\right>} \newcommand{\bra}[1]{\left<#1\right|} $ is a valid Hilbert space. But what about the physical Hilbert spaces: $$\mathcal{H}_A \odot \mathcal{H}_B=\textrm{Span}\{ \ket{i}\otimes \ket{j}+\ket{j}\otimes\ket{i}\}\tag{Bosons}$$ $$\mathcal{H}_A \wedge \mathcal{H}_B=\textrm{Span}\{ \ket{i}\otimes \ket{j}-\ket{j}\otimes\ket{i}\}\tag{Fermions}$$ How do we go about showing that they are valid Hilbert spaces (if indeed they are)?
It is trivial to show they are inner product spaces but I wouldn't know where to start in show completeness.