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.