  Stochastic integrals are defined with respect to a collection $P = (P_i; \, i
\in I)$ of continuous semimartingales, imposing no assumptions on the index set
$I$ and the subspace of $\mathbb{R}^I$ where $P$ takes values. The integrals
are constructed though finite-dimensional approximation, identifying the
appropriate local geometry that allows extension to infinite dimensions. For
local martingale integrators, the resulting space $\mathsf{S} (P)$ of
stochastic integrals has an operational characterisation via a corresponding
set of integrands $\mathsf{R} (C)$, constructed with only reference the
covariation structure $C$ of $P$. This bijection between $\mathsf{R} (C)$ and
the (closed in the semimartingale topology) set $\mathsf{S} (P)$ extends to
families of continuous semimartingale integrators for which the drift process
of $P$ belongs to $\mathsf{R} (C)$. In the context of infinite-asset models in
Mathematical Finance, the latter structural condition is equivalent to a
certain natural form of market viability. The enriched class of wealth
processes via extended stochastic integrals leads to exact analogues of
optional decomposition and hedging duality as the finite-asset case. A
corresponding characterisation of market completeness in this setting is
provided.
