An inner product space is a complex vector space with an inner product whose properties abstract those of the inner product ∑ j = 1 n z j w ¯ j in ℂ n . Given the inner product, we can define a norm, and use it to measure the lengths of the vectors in the space. It therefore makes sense to consider convergent sequences. The space is complete if and only if all Cauchy sequences converge. A Hilbert space is a complete inner product space. In infinite dimensions, convergence issues are much more delicate than many readers will be used to. In fact, it took mathematicians almost a century to go from Fourier’s initial ideas to the modern framework of Hilbert spaces. Hence this chapter will be considerably harder than the previous ones. The ideas remain fundamental to modern science.