A Hilbert space is specific example of a Banach space, because its norm comes from a scalar product. This particular norm makes the geometry of a Hilbert space very familiar to us. In particular, in a Hilbert space one can find a unique element of a closed, convex subset that minimizes the distance of this subset from a point lying outside of it. One can also think of projections of vectors on closed subspaces. Again, all this would have been impossible were the space with scalar product not complete. The chapter ends with remarkable example showing that conditional probability, one of the fundamental notions of probability theory, has much to do with projections in a Hilbert space.

Chapter 5: Many abstract concepts that make linear algebra a powerful mathematical tool have their roots in plane geometry, so we begin the study of inner product spaces with a review of basic properties of lengths and angles in the real two-dimensional plane. Guided by these geometrical properties, we formulate axioms for inner products and norms, which provide generalized notions of length (norm) and perpendicularity (orthogonality) in abstract vector spaces.