= Universe for spectra
A <universe for spectra> is a countably infinite-dimensional real <inner product space> $U$, used to index finite-dimensional suspension directions. A standard model is the algebraic <direct sum> $U=\bigoplus_{i\geq1}\mathbb R e_i$ with an <orthonormal basis>, carrying its direct-limit <topology>. Its <vectors> have finite support; this universe is not the completed <Hilbert space> $\ell^2$. Indexed <topological spectra> assign spaces to finite-dimensional <vector subspaces> $V\subseteq U$ and use the corresponding orthogonal differences for structure maps.<What precisely are E infinity ring spaces and E infinity ring spectra? by J. P. May>[Sections 2 and 5]{ref}
A family of indexing <vector subspaces> is \b[cofinal] when every finite-dimensional $W\subseteq U$ is contained in some member of the family. The flag $V_n=\operatorname{span}(e_1,\ldots,e_n)$ is cofinal: a finite <basis> of $W$ uses only finitely many of the $e_i$. In contrast, $W_n=\operatorname{span}(e_2,e_4,\ldots,e_{2n})$ has unbounded <dimension> but never contains $e_1$, so it is not cofinal. Cofinality ensures that restricting the index system does not discard any finite collection of suspension directions.
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