Students are expected to be familiar on a basic level with at least 80% of the following notions:
Vector space, linear map, subspace, quotient space, dual space, Tensor product.
Topological space, Locally compact space, metric space, Complete metric space, completion of a metric space.
Differentiable manifold, tangent space, tangent bundle.
Group, group action, abelian group,
Hilbert space, Fourier series, measure, Fourier transform