  We study Hilbert spaces $H$ interpreted, in an appropriate sense, in a
first-order theory. Under a new finiteness hypothesis that we call {\em
scatteredness} we prove that $H$ is a direct sum of {\em asymptotically free}
components, where short-range interactions are controlled by algebraic closure
and long-range interactions vanish. Examples include $L^2$-spaces relative to
Macpherson-Steinhorn definable measures; $L^2$ spaces relative to the Haar
measure of the absolute Galois groups; irreducible unitary representations of
$p$-adic Lie groups; and unitary representations of the automorphism group of
an $\omega$-categorical theory. In the last case, our main result specialises
to a theorem of Tsankov. New methods are required, making essential use of
local stability theory in continuous logic.
