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In this article, we lay foundations for a theory of infinite dimensional euclidean lattices --- and more generally, of infinite dimensional hermitian vector bundles over some "arithmetic curve" ${\rm Spec}\,\mathcal{O}_K$ attached to the ring of integers $\mathcal{O}_K$ of some number field $K$ --- with a view towards applications to transcendence theory and Diophantine geometry. In the first part of this article, we study the properties of the $\theta$-invariant $h^0_\theta(\overline{E})$ attached to some euclidean lattice $\overline{E}:= (E, \Vert.\Vert)$, defined by the expression $$h^0_\theta(\overline{E}) := \log \sum_{v \in E} e^{- \pi \Vert v \Vert^2},$$ and, more generally, of the $\theta$-invariant attached to some finite rank hermitian vector bundle $\overline{E}$ over an arithmetic curve. Then we construct categories of infinite dimensional hermitian vector bundles and we show that it is possible to associate generalized $\theta$-invariants to their objects, so that they satisfy suitable subadditivity and limit properties.