Mathematical Methods of Physics/Riesz representation theorem

In this chapter, we will more formally discuss the bra and ket notation introduced in the previous chapter. Projections Definition Let be a Hilbert space and let be a continuous linear transformation. Then is said to be a linear functional on . The space of all linear functionals on is denoted as . Notice that is a normed vector space on with We also have the obvious definition, are said to be orthogonal if . We write this as . If then we write if Theorem Let be a Hilbert space, let be a closed subspace of and let . Then, every can be written where Proof Riesz representation theorem Let be a Hilbert space. Then, every (that is is a linear functional) can be expressed as an inner product.