  We define a notion of morphism for quotient vector bundles that yields both a
category $\textit{QVBun}$ and a contravariant global sections functor
$C:\textit{QVBun}^{\textrm{op}}\to\textit{Vect}$ whose restriction to trivial
vector bundles with fiber $F$ coincides with the contravariant functor
$\textit{Top}^{\textrm{op}}\to\textit{Vect}$ of $F$-valued continuous
functions. Based on this we obtain a linear extension of the adjunction between
the categories of topological spaces and locales: (i) a linearized topological
space is a spectral vector bundle, by which is meant a mildly restricted type
of quotient vector bundle; (ii) a linearized locale is a locale $\triangle$
equipped with both a topological vector space $A$ and a $\triangle$-valued
support map for the elements of $A$ satisfying a continuity condition relative
to the spectrum of $\triangle$ and the lower Vietoris topology on
$\operatorname{Sub} A$; (iii) we obtain an adjunction between the full
subcategory of spectral vector bundles $\textit{QVBun}_{\Sigma}$ and the
category of linearized locales $\textit{LinLoc}$, which restricts to an
equivalence of categories between sober spectral vector bundles and spatial
linearized locales. The spectral vector bundles are classified by a finer
topology on $\operatorname{Sub} A$, called the open support topology, but there
is no notion of universal spectral vector bundle for an arbitrary topological
vector space $A$.
