We study examples of set-theoretic phenomena occurring in infinite-dimensional spaces, motivated by functional analysis. This includes equivalence relations induced by ideals of operators on a Hilbert space, a new "local" Ramsey theory for block sequences in Banach spaces and countable discrete vector spaces, analogues of selective ultrafilters and coideals in these settings, and families of infinite-dimensional subspaces which have pairwise finite-dimensional intersection. We draw analogies to the structure of the infinite subsets of the natural numbers.