  We carry out a systematic study of decidability for theories of (a) real
vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces,
Banach spaces and metric spaces, all formalised using a 2-sorted first-order
language. The theories for list (a) turn out to be decidable while the theories
for list (b) are not even arithmetical: the theory of 2-dimensional Banach
spaces, for example, has the same many-one degree as the set of truths of
second-order arithmetic.
  We find that the purely universal and purely existential fragments of the
theory of normed spaces are decidable, as is the AE fragment of the theory of
metric spaces. These results are sharp of their type: reductions of Hilbert's
10th problem show that the EA fragments for metric and normed spaces and the AE
fragment for normed spaces are all undecidable.
